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Basic
TECHNICAL MATHEMATICS with CALCULUS
Allyn J. Washington Richard S. Evans
Eleventh Edition
ELEVENTH EDITION
Basic Technical Mathematics with Calculus
Allyn J.Washington
Dutchess Community College
Richard S. Evans
Corning Community College
Director, Courseware Portfolio Management: Deirdre Lynch Executive Editor: Jeff Weidenaar Editorial Assistant: Jennifer Snyder Managing Producer: Karen Wernholm Content Producer: Tamela Ambush Producer: Jean Choe Manager, Content Development: Kristina Evans Math Content Developer: Megan M. Burns Field Marketing Manager: Jennifer Crum Product Marketing Manager: Alicia Frankel Marketing Assistant: Hanna Lafferty Senior Author Support/Technology Specialist: Joe Vetere Manager, Rights Management: Gina M. Cheselka Manufacturing Buyer: Carol Melville, LSC Communications Composition: SPi Global Associate Director of Design: Blair Brown Cover Design: Barbara Atkinson Cover Image: Daniel Schoenen/Image Broker/Alamy Stock Photo
Copyright © 2018, 2014, 2009 by Pearson Education, Inc. or its affiliates. All Rights Reserved. Printed in the United States of America. This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise. For information regarding permissions, request forms and the appropriate contacts within the Pearson Rights & Permissions Department, please visit www.pearsoned.com/permissions/.
PEARSON, ALWAYS LEARNING, and MYMATHLAB are exclusive trademarks owned by Pearson Education, Inc. or its affiliates in the United States and/or other countries.
Unless otherwise indicated herein, any third-party trademarks that may appear in this work are the property of their respective owners and any references to third-party trademarks, logos or other trade dress are for demonstrative or descriptive purposes only. Such references are not intended to imply any sponsorship, endorsement, authorization, or promotion of Pearson’s products by the owners of such marks, or any relationship between the owner and Pearson Education, Inc. or its affiliates, authors, licensees or distributors.
Library of Congress Cataloging-in-Publication Data
Names: Washington, Allyn J. | Evans, Richard (Mathematics teacher) Title: Basic technical mathematics with calculus / Allyn J. Washington, Dutchess Community
College, Richard Evans, Corning Community College. Description: 11th edition. | Boston : Pearson, [2018] | Includes indexes. Identifiers: LCCN 2016020426| ISBN 9780134437736 (hardcover) |
ISBN 013443773X (hardcover) Subjects: LCSH: Mathematics–Textbooks. | Calculus–Textbooks. Classification: LCC QA37.3 .W38 2018 | DDC 510–dc23 LC record available at https://lccn.loc.gov/2016020426
1 16
Student Edition: ISBN 10: 0-13-443773-X ISBN 13: 978-0-13-443773-6
Contents
Preface
VII
1 Basic Algebraic Operations
1
1.1 Numbers
2
1.2 Fundamental Operations of Algebra
6
1.3 Calculators and Approximate Numbers
12
1.4 Exponents and Unit Conversions
17
1.5 Scientific Notation
24
1.6 Roots and Radicals
27
1.7 Addition and Subtraction of Algebraic
Expressions
29
1.8 Multiplication of Algebraic Expressions
33
1.9 Division of Algebraic Expressions
36
1.10 Solving Equations
39
1.11 Formulas and Literal Equations
43
1.12 Applied Word Problems
46
Key Formulas and Equations, Review Exercises,
and Practice Test
50
2 Geometry
54
2.1 Lines and Angles
55
2.2 Triangles
58
2.3 Quadrilaterals
65
2.4 Circles
68
2.5 Measurement of Irregular Areas
72
2.6 Solid Geometric Figures
76
Key Formulas and Equations, Review Exercises,
and Practice Test
80
3 Functions and Graphs
85
3.1 Introduction to Functions
86
3.2 More about Functions
89
3.3 Rectangular Coordinates
94
3.4 The Graph of a Function
96
3.5 Graphs on the Graphing Calculator
102
3.6 Graphs of Functions Defined by
Tables of Data
107
Review Exercises and Practice Test
110
4 The Trigonometric Functions
113
4.1 Angles
114
4.2 Defining the Trigonometric Functions
117
4.3 Values of the Trigonometric Functions
120
4.4 The Right Triangle
124
4.5 Applications of Right Triangles
129
Key Formulas and Equations, Review Exercises,
and Practice Test
134
5 Systems of Linear Equations;
Determinants
140
5.1 Linear Equations and Graphs of Linear
Functions
141
5.2 Systems of Equations and Graphical
Solutions
147
5.3 Solving Systems of Two Linear Equations
in Two Unknowns Algebraically
152
5.4 Solving Systems of Two Linear Equations
in Two Unknowns by Determinants
159
5.5 Solving Systems of Three Linear Equations in
Three Unknowns Algebraically
164
5.6 Solving Systems of Three Linear Equations in
Three Unknowns by Determinants
169
Key Formulas and Equations, Review Exercises,
and Practice Test
174
6 Factoring and Fractions
180
6.1 Factoring: Greatest Common Factor and
Difference of Squares
181
6.2 Factoring Trinomials
186
6.3 The Sum and Difference of Cubes
193
6.4 Equivalent Fractions
195
6.5 Multiplication and Division of Fractions 200
6.6 Addition and Subtraction of Fractions
204
6.7 Equations Involving Fractions
210
Key Formulas and Equations, Review Exercises,
and Practice Test
215
7 Quadratic Equations
219
7.1 Quadratic Equations; Solution by Factoring 220
7.2 Completing the Square
225
7.3 The Quadratic Formula
227
7.4 The Graph of the Quadratic Function
232
Key Formulas and Equations, Review Exercises,
and Practice Test
237
iii
iv
ConTEnTs
8 Trigonometric Functions
of Any Angle
240
8.1 Signs of the Trigonometric Functions
241
8.2 Trigonometric Functions of Any Angle
243
8.3 Radians
249
8.4 Applications of Radian Measure
253
Key Formulas and Equations, Review Exercises,
and Practice Test
259
9 Vectors and Oblique Triangles 263
9.1 Introduction to Vectors
264
9.2 Components of Vectors
268
9.3 Vector Addition by Components
272
9.4 Applications of Vectors
277
9.5 Oblique Triangles, the Law of Sines
283
9.6 The Law of Cosines
290
Key Formulas and Equations, Review Exercises,
and Practice Test
295
10 Graphs of the Trigonometric
Functions
299
10.1 Graphs of y 5 a sin x and y 5 a cos x
300
10.2 Graphs of y 5 a sin bx and y 5 a cos bx
303
10.3 Graphs of y 5 a sin (bx 1 c) and
y 5 a cos (bx 1 c)
306
10.4 Graphs of y 5 tan x, y 5 cot x, y 5 sec x,
y 5 csc x
310
10.5 Applications of the Trigonometric Graphs 312
10.6 Composite Trigonometric Curves
315
Key Formulas and Equations, Review Exercises,
and Practice Test
320
11 Exponents and Radicals
323
11.1 Simplifying Expressions with Integer
Exponents
324
11.2 Fractional Exponents
328
11.3 Simplest Radical Form
332
11.4 Addition and Subtraction of Radicals
336
11.5 Multiplication and Division of Radicals
338
Key Formulas and Equations, Review Exercises,
and Practice Test
342
12 Complex Numbers
12.1 Basic Definitions 12.2 Basic Operations with Complex Numbers 12.3 Graphical Representation of Complex
Numbers
345
346 349
352
12.4 Polar Form of a Complex Number
354
12.5 Exponential Form of a Complex Number 356
12.6 Products, Quotients, Powers, and Roots of
Complex Numbers
358
12.7 An Application to Alternating-current (ac)
Circuits
364
Key Formulas and Equations, Review Exercises,
and Practice Test
370
13 Exponential and Logarithmic
Functions
373
13.1 Exponential Functions
374
13.2 Logarithmic Functions
376
13.3 Properties of Logarithms
380
13.4 Logarithms to the Base 10
385
13.5 Natural Logarithms
388
13.6 Exponential and Logarithmic Equations 391
13.7 Graphs on Logarithmic and
Semilogarithmic Paper
395
Key Formulas and Equations, Review Exercises,
and Practice Test
400
14 Additional Types of Equations
and Systems of Equations
403
14.1 Graphical Solution of Systems of Equations 404
14.2 Algebraic Solution of Systems of Equations 407
14.3 Equations in Quadratic Form
411
14.4 Equations with Radicals
414
Review Exercises and Practice Test
418
15 Equations of Higher Degree
420
15.1 The Remainder and Factor
Theorems; Synthetic Division
421
15.2 The Roots of an Equation
426
15.3 Rational and Irrational Roots
431
Key Formulas and Equations, Review Exercises,
and Practice Test
436
16 Matrices; Systems of Linear
Equations
439
16.1 Matrices: Definitions and Basic Operations 440
16.2 Multiplication of Matrices
444
16.3 Finding the Inverse of a Matrix
449
16.4 Matrices and Linear Equations
453
16.5 Gaussian Elimination
457
16.6 Higher-order Determinants
461
Key Formulas and Equations, Review Exercises,
and Practice Test
466
ConTEnTs
v
17 Inequalities
470
17.1 Properties of Inequalities
471
17.2 Solving Linear Inequalities
475
17.3 Solving Nonlinear Inequalities
480
17.4 Inequalities Involving Absolute Values
486
17.5 Graphical Solution of Inequalities
with Two Variables
489
17.6 Linear Programming
492
Key Formulas and Equations, Review Exercises,
and Practice Test
496
18 Variation
499
18.1 Ratio and Proportion
500
18.2 Variation
504
Key Formulas and Equations, Review Exercises,
and Practice Test
510
19 Sequences and the Binomial
Theorem
514
19.1 Arithmetic Sequences
515
19.2 Geometric Sequences
519
19.3 Infinite Geometric Series
522
19.4 The Binomial Theorem
526
Key Formulas and Equations, Review Exercises,
and Practice Test
531
20 Additional Topics in Trigonometry 535
20.1 Fundamental Trigonometric Identities
536
20.2 The Sum and Difference Formulas
542
20.3 Double-Angle Formulas
547
20.4 Half-Angle Formulas
551
20.5 Solving Trigonometric Equations
554
20.6 The Inverse Trigonometric Functions
558
Key Formulas and Equations, Review Exercises,
and Practice Test
564
21 Plane Analytic Geometry
21.1 Basic Definitions 21.2 The Straight Line 21.3 The Circle 21.4 The Parabola 21.5 The Ellipse 21.6 The Hyperbola 21.7 Translation of Axes 21.8 The Second-degree Equation 21.9 Rotation of Axes
568
569 573 579 584 588 593 599 602 605
21.10 Polar Coordinates
609
21.11 Curves in Polar Coordinates
612
Key Formulas and Equations, Review Exercises,
and Practice Test
616
22 Introduction to Statistics
621
22.1 Graphical Displays of Data
622
22.2 Measures of Central Tendency
626
22.3 Standard Deviation
630
22.4 Normal Distributions
633
22.5 Statistical Process Control
637
22.6 Linear Regression
642
22.7 Nonlinear Regression
647
Key Formulas and Equations, Review Exercises,
and Practice Test
650
23 The Derivative
655
23.1 Limits
656
23.2 The Slope of a Tangent to a Curve
664
23.3 The Derivative
667
23.4 The Derivative as an Instantaneous Rate of
Change
671
23.5 Derivatives of Polynomials
675
23.6 Derivatives of Products and Quotients of
Functions
680
23.7 The Derivative of a Power of a Function 684
23.8 Differentiation of Implicit Functions
690
23.9 Higher Derivatives
693
Key Formulas and Equations, Review Exercises,
Practice Test
696
24 Applications of the Derivative 700
24.1 Tangents and Normals
701
24.2 Newton’s Method for Solving Equations 703
24.3 Curvilinear Motion
706
24.4 Related Rates
711
24.5 Using Derivatives in Curve Sketching
715
24.6 More on Curve Sketching
721
24.7 Applied Maximum and Minimum Problems 726
24.8 Differentials and Linear Approximations 733
Key Formulas and Equations, Review Exercises,
Practice Test
737
25 Integration
25.1 Antiderivatives 25.2 The Indefinite Integral 25.3 The Area Under a Curve
742
743 745 750
vi
ConTEnTs
25.4 The Definite Integral
755
25.5 Numerical Integration:
The Trapezoidal Rule
758
25.6 Simpson's Rule
761
Key Formulas and Equations, Review Exercises,
Practice Test
765
26 Applications of Integration
768
26.1 Applications of the Indefinite Integral
769
26.2 Areas by Integration
773
26.3 Volumes by Integration
779
26.4 Centroids
784
26.5 Moments of Inertia
790
26.6 Other Applications
795
Key Formulas and Equations, Review Exercises,
Practice Test
800
27 Differentiation of Transcendental
Functions
805
27.1 Derivatives of the Sine and Cosine
Functions
806
27.2 Derivatives of the Other Trigonometric
Functions
810
27.3 Derivatives of the Inverse Trigonometric
Functions
813
27.4 Applications
816
27.5 Derivative of the Logarithmic Function
821
27.6 Derivative of the Exponential Function
825
27.7 L’Hospital’s Rule
828
27.8 Applications
832
Key Formulas and Equations, Review Exercises,
Practice Test
835
28 Methods of Integration
840
28.1 The Power Rule for Integration
841
28.2 The Basic Logarithmic Form
843
28.3 The Exponential Form
847
28.4 Basic Trigonometric Forms
850
28.5 Other Trigonometric Forms
854
28.6 Inverse Trigonometric Forms
858
28.7 Integration by Parts
862
28.8 Integration by Trigonometric Substitution 866
28.9 Integration by Partial Fractions:
Nonrepeated Linear Factors
869
28.10 Integration by Partial Fractions:
Other Cases
872
28.11 Integration by Use of Tables
877
Key Formulas and Equations, Review Exercises,
Practice Test
880
29 Partial Derivatives and Double
Integrals
884
29.1 Functions of Two Variables
885
29.2 Curves and Surfaces in Three Dimensions 888
29.3 Partial Derivatives
894
29.4 Double Integrals
898
Key Formulas and Equations, Review Exercises,
Practice Test
902
30 Expansion of Functions in Series 904
30.1 Infinite Series
905
30.2 Maclaurin Series
909
30.3 Operations with Series
913
30.4 Computations by Use of Series Expansions 917
30.5 Taylor Series
920
30.6 Introduction to Fourier Series
923
30.7 More About Fourier Series
928
Key Formulas and Equations, Review Exercises,
Practice Test
933
31 Differential Equations
937
31.1 Solutions of Differential Equations
938
31.2 Separation of Variables
940
31.3 Integrating Combinations
943
31.4 The Linear Differential Equation
of the First Order
946
31.5 Numerical Solutions of First-order
Equations
948
31.6 Elementary Applications
951
31.7 Higher-order Homogeneous Equations 957
31.8 Auxiliary Equation with Repeated
or Complex Roots
961
31.9 Solutions of Nonhomogeneous Equations 964
31.10 Applications of Higher-order Equations 969
31.11 Laplace Transforms
976
31.12 Solving Differential Equations by Laplace
Transforms
981
Key Formulas and Equations, Review Exercises,
Practice Test
985
Appendix A Solving Word Problems
A.1
Appendix B Units of Measurement
A.2
Appendix C Newton’s Method
A.4
Appendix D A Table of Integrals
A.5
Photo Credits
A.8
Answers to Odd-Numbered Exercises
and Chapter Review Exercises
B.1
Solutions to Practice Test Problems
C.1
Index of Applications
D.1
Index
E.1
Preface
scope of the Book
Basic Technical Mathematics with Calculus, Eleventh Edition, is intended primarily for students in technical and pre-engineering technical programs or other programs for which coverage of mathematics is required. Chapters 1 through 20 provide the necessary background for further study with an integrated treatment of algebra and trigonometry. Chapter 21 covers the basic topics of analytic geometry, and Chapter 22 gives an introduction to statistics. Chapters 23 through 31 cover fundamental concepts of calculus including limits, derivatives, integrals, series representation of functions, and differential equations. In the examples and exercises, numerous applications from the various fields of technology are included, primarily to indicate where and how mathematical techniques are used. However, it is not necessary that the student have a specific knowledge of the technical area from which any given problem is taken. Most students using this text will have a background that includes some algebra and geometry. However, the material is presented in adequate detail for those who may need more study in these areas. The material presented here is sufficient for two to three semesters. One of the principal reasons for the arrangement of topics in this text is to present material in an order that allows a student to take courses concurrently in allied technical areas, such as physics and electricity. These allied courses normally require a student to know certain mathematics topics by certain definite times; yet the traditional order of topics in mathematics courses makes it difficult to attain this coverage without loss of continuity. However, the material in this book can be rearranged to fit any appropriate sequence of topics. The approach used in this text is not unduly rigorous mathematically, although all appropriate terms and concepts are introduced as needed and given an intuitive or algebraic foundation. The aim is to help the student develop an understanding of mathematical methods without simply providing a collection of formulas. The text material is developed recognizing that it is essential for the student to have a sound background in algebra and trigonometry in order to understand and succeed in any subsequent work in mathematics.
new to This Edition
You may have noticed something new on the cover of this book. Another author! Yes, after 50 years as a “solo act,” Allyn Washington has a partner. New co-author Rich Evans is a veteran faculty member at Corning Community College (NY) and has brought a wealth of positive contributions to the book and accompanying MyMathLab course.
The new features of the eleventh edition include:
CAUTION When you enter URLs for the Graphing Calculator Manual, take care to distinguish the following characters: l = lowercase l I = uppercase I
1 = one O = uppercase O
0 = zero ■
• Refreshed design – The book has been redesigned in full color to help students better use it and to help motivate students as they put in the hard work to learn the mathematics (because let’s face it—a more modern looking book has more appeal).
• Graphing calculator – We have replaced the older TI-84 screens with those from the new TI-84 Plus-C (the color version). And Benjamin Rushing [Northwestern State University] has added graphing calculator help for students, accessible online via short URLs in the margins. If you’d like to see the complete listing of entries for the online graphing calculator manual, use the URL goo.gl/eAUgW3.
• Applications – The text features a wealth of new applications in the examples and exercises (over 200 in all!). Here is a sampling of the contexts for these new applications:
Power of a wind turbine (Section 3.4)
Height of One World Trade Center (Section 4.4)
GPS satellite velocity (Section 8.4)
Google’s self-driving car laser distance (Section 9.6)
Phase angle for current/voltage lead and lag (Section 10.3)
Growth of computer processor transistor counts (Section 13.7)
vii
viii PREFaCE
Bezier curve roof design (Section 15.3)
Cardioid microphone polar pattern (Section 21.7)
Social networks usage (Section 22.1)
Video game system market share (Section 22.1)
Bluetooth headphone maximum revenue (Section 24.7)
Saddledome roof slopes (Section 29.3)
Weight loss differential equation (Section 31.6)
• Exercises – There are over 1000 new and updated exercises in the new edition. In creating new exercises, the authors analyzed aggregated student usage and performance data from MyMathLab for the previous edition of this text. The results of this analysis helped improve the quality and quantity of exercises that matter the most to instructors and students. There are a total of 14,000 exercises and 1400 examples in the eleventh edition.
• Chapter Endmatter – The exercises formerly called “Quick Chapter Review” are now labeled “Concept Check Exercises” (to better communicate their function within the chapter endmatter).
• MyMathLab – Features of the MyMathLab course for the new edition include: Hundreds of new assignable algorithmic exercises help you address the homework needs of students. Additionally, all exercises are in the new HTML5 player, so they are accessible via mobile devices.
223 new instructional videos (to augment the existing 203 videos) provide help for students as they do homework. These videos were created by Sue Glascoe (Mesa Community College) and Benjamin Rushing (Northwestern State University).
A new Graphing Calculator Manual, created specifically for this text, features instructions for the TI-84 and TI-89 family of calculators.
New PowerPoint® files feature animations that are designed to help you better teach key concepts.
Study skills modules help students with the life skills (e.g., time management) that can make the difference between passing and failing. Content updates for the eleventh edition were informed by the extensive reviews of the text completed for this revision. These include:
• Unit analysis, including operations with units and unit conversions, has been moved from Appendix B to Section 1.4. Appendix B has been streamlined, but still contains the essential reference materials on units.
• In Section 1.3, more specific instructions have been provided for rounding combined operations with approximate numbers.
• Engineering notation has been added to Section 1.5. • Finding the domain and range of a function graphically has been added to Section 3.4. • The terms input, output, piecewise defined functions, and practical domain and range
have been added to Chapter 3.
• In response to reviewer feedback, the beginning of Chapter 5 has been reorganized so that systems of equations has a strong introduction in Section 5.2. The prerequisite material needed for systems of equations (linear equations and graphs of linear functions) has been consolidated into Section 5.1. An example involving linear regression has also been added to Section 5.1.
• Solving systems using reduced row echelon form (rref) on a calculator has been added to Chapter 5.
• Several reviewers made the excellent suggestion to strengthen the focus on factoring in Chapter 6 by taking the contents of 6.1 (Special Products) and spreading it throughout the chapter. This change has been implemented. The terminology greatest common factor (GCF) has also been added to this chapter.
Continuing Features
PREFaCE
ix
• In Chapter 7, the square root property is explicitly stated and illustrated. • In Chapter 8, the unit circle definition of the trigonometric functions has been added. • In Chapter 9, more emphasis had been given to solving equilibrium problems, includ-
ing those that have more than one unknown.
• In Chapter 10, an example was added to show how the phase angle can be interpreted, and how it is different from the phase shift.
• In Chapter 16, the terminology row echelon form is used. Also, solving a system using rref is again illustrated. The material on using properties to evaluate determinants was deleted.
• The terminology binomial coefficients was added to Chapter 19. • Chapter 22 (Introduction to Statistics) has undergone significant changes.
Section 22.1 now discusses common graphs used for both qualitative data (bar graphs and pie charts) and quantitative data (histograms, stem-and-leaf plots, and time series plots).
In Section 22.2, what was previously called the arithmetic mean is now referred to as simply the mean.
The empirical rule had been added to Section 22.4.
The sampling distribution of x has been formalized including the statement of the central limit theorem.
A discussion of interpolation and extrapolation has been added in the context of regression, as well as information on how to interpret the values of r and r2.
The emphasis of Section 22.7 on nonlinear regression has been changed. Information on how to choose an appropriate type of model depending on the shape of the data has been added. However, a calculator is now used to obtain the actual regression equation.
• In Chapter 23, the terminology direct substitution has been introduced in the context of limits.
• Throughout the calculus chapters, many of the differentiation and integration rules have been given names so they can be easily referred to. These include, the constant rule, power rule, constant multiple rule, product rule, quotient rule, general power rule, power rule for integration, etc.
• In Chapter 30, the proof of the Fourier coefficients has been moved online.
PagE LayouT
Special attention has been given to the page layout. We specifically tried to avoid breaking examples or important discussions across pages. Also, all figures are shown immediately adjacent to the material in which they are discussed. Finally, we tried to avoid referring to equations or formulas by number when the referent is not on the same page spread.
ChaPTER inTRoduCTions
Each chapter introduction illustrates specific examples of how the development of technology has been related to the development of mathematics. In these introductions, it is shown that these past discoveries in technology led to some of the methods in mathematics, whereas in other cases mathematical topics already known were later very useful in bringing about advances in technology. Also, each chapter introduction contains a photo that refers to an example that is presented within that chapter.
x
PREFaCE
WoRKEd-ouT ExamPLEs
subscripts l -
and wn
E X A M P L E 3 symbol in capital and in lowercase—forces on a beam
L1wL + 2P2
In the study of the forces on a certain beam, the equation W =
is used.
8
Solve for P.
8L1wL + 2P2 8W =
8 8W = L1wL + 2P2
multiply both sides by 8 simplify right side
8W = wL2 + 2LP
remove parentheses
8W - wL2 = 2LP
subtract wL2 from both sides
8W - wL2
P=
divide both sides by 2L and switch sides
■
2L
• “HELP TEXT” Throughout the book, special explanatory comments in blue type have been used in the examples to emphasize and clarify certain important points. Arrows are often used to indicate clearly the part of the example to which reference is made.
• EXAMPLE DESCRIPTIONS A brief descriptive title is given for each example. This gives an easy reference for the example, particularly when reviewing the contents of the section.
• APPLICATION PROBLEMS There are over 350 applied examples throughout the text that show complete solutions of application problems. Many relate to modern technology such as computer design, electronics, solar energy, lasers, fiber optics, the environment, and space technology. Others examples and exercises relate to technologies such as aeronautics, architecture, automotive, business, chemical, civil, construction, energy, environmental, fire science, machine, medical, meteorology, navigation, police, refrigeration, seismology, and wastewater. The Index of Applications at the end of the book shows the breadth of applications in the text.
KEy FoRmuLas and PRoCEduREs
Throughout the book, important formulas are set off and displayed so that they can be easily referenced for use. Similarly, summaries of techniques and procedures consistently appear in color-shaded boxes.
noTE →
“CauTion” and “noTE” indiCaToRs
CAUTION This heading is used to identify errors students commonly make or places where they frequently have difficulty. ■
The NOTE label in the side margin, along with accompanying blue brackets in the main body of the text, points out material that is of particular importance in developing or
understanding the topic under discussion. [Both of these features have been clarified in
the eleventh edition by adding a small design element to show where the CAUTION or
NOTE feature ends.]
ChaPTER and sECTion ConTEnTs
A listing of learning outcomes for each chapter is given on the introductory page of the chapter. Also, a listing of the key topics of each section is given below the section number and title on the first page of the section. This gives the student and instructor a quick preview of the chapter and section contents.
PRaCTiCE ExERCisEs
Most sections include some practice exercises in the margin. They are included so that a student is more actively involved in the learning process and can check his or her understanding of the material. They can also be used for classroom exercises. The answers to these exercises are given at the end of the exercises set for the section. There are over 450 of these exercises.
FEaTuREs oF ExERCisEs
• EXERCISES DIRECTLY REFERENCED TO TEXT EXAMPLES The first few exercises in most of the text sections are referenced directly to a specific example of the section. These exercises are worded so that it is necessary for the student to refer to the example in order to complete the required solution. In this way, the student should be able to better review and understand the text material before attempting to solve the exercises that follow.
Technology and supplements
PREFaCE
xi
• WRITING EXERCISES There are over 270 writing exercises through the book (at least eight in each chapter) that require at least a sentence or two of explanation as part of the answer. These are noted by a pencil icon next to the exercise number.
• APPLICATION PROBLEMS There are about 3000 application exercises in the text that represent the breadth of applications that students will encounter in their chosen professions. The Index of Applications at the end of the book shows the breadth of applications in the text.
ChaPTER EndmaTTER • KEY FORMULAS AND EQUATIONS Here all important formulas and equations
are listed together with their corresponding equation numbers for easy reference.
• CHAPTER REVIEW EXERCISES These exercises consist of (a) Concept Check Exercises (a set of true/false exercises) and (b) Practice and Applications.
• CHAPTER TEST These are designed to mirror what students might see on the actual chapter test. Complete step-by-step solutions to all practice test problems are given in the back of the book.
maRgin noTEs Throughout the text, some margin notes point out relevant historical events in mathematics and technology. Other margin notes are used to make specific comments related to the text material. Also, where appropriate, equations from earlier material are shown for reference in the margin.
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PREFaCE xiii
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xiv PREFaCE
acknowledgments
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sTudEnT’s soLuTions manuaL ISBN-10: 0134434633 | ISBN-13: 9780134434636
The Student’s Solutions Manual by Matthew Hudelson (Washington State University) includes detailed solutions for every other odd-numbered section exercise. The manual is available in print and is downloadable from within MyMathLab.
insTRuCToR’s soLuTions manuaL (doWnLoadaBLE) ISBN-10: 0134435893 | ISBN-13: 9780134435893
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TEsTgEn (doWnLoadaBLE) ISBN-10: 0134435753 | ISBN-13: 9780134435756
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Special thanks goes to Matthew Hudelson of Washington State University for preparing the Student’s Solutions Manual and the Instructor’s Solutions Manual. Thanks also to Bob Martin and John Garlow, both of Tarrant County College (TX) for their work on these manuals for previous editions. A special thanks to Ben Rushing of Northwestern State University of Louisiana for his work on the graphing calculator manual as well as instructional videos. Our gratitude is also extended to to Sue Glascoe (Mesa Community College) for creating instructional videos. We would also like to express appreciation for the work done by David Dubriske and Cindy Trimble in checking for accuracy in the text and exercises. Also, we again wish to thank Thomas Stark of Cincinnati State Technical and Community College for the RISERS approach to solving word problems in Appendix A. We also extend our thanks to Julie Hoffman, Personal Assistant to Allyn Washington.
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xv
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Finally, we wish to sincerely thank again each of the over 375 reviewers of the eleven editions of this text. Their comments have helped further the education of more than two million students during since this text was first published in 1964.
Allyn Washington
Richard Evans
Basic Algebraic Operations
1
Interest in things such as the land on which they lived, the structures they built, and the motion of the planets led people in early civilizations to keep records and to create methods of counting and measuring.
In turn, some of the early ideas of arithmetic, geometry, and trigonometry were developed. From such beginnings, mathematics has played a key role in the great advances in science and technology.
Often, mathematical methods were developed from scientific studies made in particular areas, such as astronomy and physics. Many people were interested in the math itself and added to what was then known. Although this additional mathematical knowledge may not have been related to applications at the time it was developed, it often later became useful in applied areas.
In the chapter introductions that follow, examples of the interaction of technology and mathematics are given. From these examples and the text material, it is hoped you will better understand the important role that math has had and still has in technology. In this text, there are applications from technologies including (but not limited to) aeronautical, business, communications, electricity, electronics, engineering, environmental, heat and air conditioning, mechanical, medical, meteorology, petroleum, product design, solar, and space.
We begin by reviewing the concepts that deal with numbers and symbols. This will enable us to develop topics in algebra, an understanding of which is essential for progress in other areas such as geometry, trigonometry, and calculus.
LEARNING OUTCOMES
After completion of this chapter, the student should be able to:
• Identify real, imaginary, rational, and irrational numbers
• Perform mathematical operations on integers, decimals, fractions, and radicals
• Use the fundamental laws of algebra in numeric and algebraic expressions
• Employ mathematical order of operations
• Understand technical measurement, approximation, the use of significant digits, and rounding
• Use scientific and engineering notations • Convert units of measurement • Rearrange and solve basic algebraic
equations • Interpret word problems using
algebraic symbols
◀ From the great Pyramid of giza, built in Egypt 4500 years ago, to the modern technology of today, mathematics has played a key role in the advancement of civilization. along the way, important discoveries have been made in areas such as architecture, navigation, transportation, electronics, communication, and astronomy. mathematics will continue to pave the way for new discoveries.
1
2
ChaPTER 1 Basic Algebraic Operations
1.1 Numbers
Real Number System • Number Line • Absolute Value • Signs of Inequality • Reciprocal • Denominate Numbers • Literal Numbers
■ Irrational numbers were discussed by the Greek mathematician Pythagoras in about 540 B.C.E.
In technology and science, as well as in everyday life, we use the very familiar counting
numbers, or natural numbers 1, 2, 3, and so on. The whole numbers include 0 as well
as all the natural numbers. Because it is necessary and useful to use negative numbers as
well as positive numbers in mathematics and its applications, the natural numbers are called the positive integers, and the numbers -1, -2, -3, and so on are the negative integers.
Therefore, the integers include the positive integers, the negative integers, and zero,
which is neither positive nor negative. This means that the integers are the numbers . . . , -3, -2, -1, 0, 1, 2, 3, . . . and so on.
A rational number is a number that can be expressed as the division of one integer a by another nonzero integer b, and can be represented by the fraction a>b. Here a is the numerator and b is the denominator. Here we have used algebra by letting letters rep-
resent numbers.
Another type of number, an irrational number, cannot be written in the form of a
fraction that is the division of one integer by another integer. The following example
illustrates integers, rational numbers, and irrational numbers.
■ For reference, p = 3.14159265 c
■ A notation that is often used for repeating
decimals is to place a bar over the digits that
repeat. Using this notation we can write
1121 1665
=
0.6732
and
2 3
=
0.6.
E X A M P L E 1 Identifying rational numbers and irrational numbers
The numbers 5 and -19 are integers. They are also rational numbers because they can be
written
as
5 1
and
-119,
respectively.
Normally,
we
do
not
write
the
1’s
in
the
denominators.
The
numbers
5 8
and
- 11 3
are
rational
numbers
because
the
numerator
and
the
denomina-
tor of each are integers.
The numbers 22 and p are irrational numbers. It is not possible to find two integers,
one divided by the other, to represent either of these numbers. In decimal form, irrational
numbers are nonterminating, nonrepeating decimals. It can be shown that square roots
(and other roots) that cannot be expressed exactly in decimal form are irrational. Also,
22 7
is
sometimes
used
as
an
approximation
for
p,
but
it
is
not
equal
exactly
to
p.
We
must
remember
that
22 7
is
rational
and
p
is
irrational.
The decimal number 1.5 is rational since it can be written as 32. Any such terminating
decimal is rational. The number 0.6666 . . . , where the 6’s continue on indefinitely, is
rational because we may write it as 23. In fact, any repeating decimal (in decimal form, a
specific sequence of digits is repeated indefinitely) is rational. The decimal number
0.6732732732 . . . is a repeating decimal where the sequence of digits 732 is repeated
indefinitely 10.6732732732 c = 111626152.
■
The rational numbers together with the irrational numbers, including all such numbers that are positive, negative, or zero, make up the real number system (see Fig. 1.1). There are times we will encounter an imaginary number, the name given to the square root of a
Imaginary 1-4, 1-7
-
3 8
Rational
1.6-
Real Numbers
4.72 Integers
Irrational p, 13, 15
Whole
Natural ... -3, -2, -1 0 1, 2, 3, ...
5 9
Fig. 1.1
1.1 Numbers
3
negative number. Imaginary numbers are not real numbers and will be discussed in Chapter 12. However, unless specifically noted, we will use real numbers. Until Chapter 12, it will be necessary to only recognize imaginary numbers when they occur.
Also in Chapter 12, we will consider complex numbers, which include both the real numbers and imaginary numbers. See Exercise 39 of this section.
■ Real numbers and imaginary numbers are both included in the complex number
system. See Exercise 39.
E X A M P L E 2 Identifying real numbers and imaginary numbers
(a) The number 7 is an integer. It is also rational because 7 = 17, and it is a real number since the real numbers include all the rational numbers.
(b) The number 3p is irrational, and it is real because the real numbers include all the
irrational numbers.
(c) The numbers 2 -10 and - 2 -7 are imaginary numbers.
(d)
The number
-3 7
is
rational
and
real.
The
number
- 27 is irrational and real.
(e)
The
number
p 6
is
irrational
and
real.
The
number
2-3 2
is
imaginary.
■
■ Fractions were used by early Egyptians and Babylonians. They were used for calculations that involved parts of measurements, property, and possessions.
A fraction may contain any number or symbol representing a number in its numerator or in its denominator. The fraction indicates the division of the numerator by the denominator, as we previously indicated in writing rational numbers. Therefore, a fraction may be a number that is rational, irrational, or imaginary.
E X A M P L E 3 Fractions
(a)
The
numbers
2 7
and
-3 2
are
fractions,
and
they
are
rational.
(b)
The
numbers
22 9
and
6 p
are
fractions,
but
they
are
not
rational
numbers.
It
is
not
possible to express either as one integer divided by another integer.
(c)
The
number
2-5 6
is
a
fraction,
and
it
is
an
imaginary
number.
■
THE NUMBER LINE
Real numbers may be represented by points on a line. We draw a horizontal line and designate some point on it by O, which we call the origin (see Fig. 1.2). The integer zero is located at this point. Equal intervals are marked to the right of the origin, and the positive integers are placed at these positions. The other positive rational numbers are located between the integers. The points that cannot be defined as rational numbers represent irrational numbers. We cannot tell whether a given point represents a rational number or an irrational number unless it is specifically marked to indicate its value.
-
26 5
-111
-p 2
4
19
9
1.7
p
4
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Negative direction
Origin Fig. 1.2
Positive direction
The negative numbers are located on the number line by starting at the origin and marking off equal intervals to the left, which is the negative direction. As shown in Fig. 1.2, the positive numbers are to the right of the origin and the negative numbers are to the left of the origin. Representing numbers in this way is especially useful for graphical methods.
4
ChaPTER 1 Basic Algebraic Operations
We next define another important concept of a number. The absolute value of a positive number is the number itself, and the absolute value of a negative number is the corresponding positive number. On the number line, we may interpret the absolute value of a number as the distance (which is always positive) between the origin and the number. Absolute value is denoted by writing the number between vertical lines, as shown in the following example.
E X A M P L E 4 absolute value
The absolute value of 6 is 6, and the absolute value of -7 is 7. We write these as 0 6 0 = 6 and 0 -7 0 = 7. See Fig. 1.3.
ƒ-7ƒ = 7 7 units
ƒ6ƒ = 6 6 units
Practice Exercises
1. 0 -4.2 0 = ?
2.
-
`
-
3 4
`
=?
■ The symbols = , 6, and 7 were introduced by English mathematicians in the late 1500s.
-8
-4
0
4
8
Fig. 1.3
Other examples are
0
7 5
0=
75,
0 - 22 0
=
22,
000
=
0,
- 0p0
=
- p,
0 -5.29 0
=
5.29,
and - 0 -9 0 = -9 since 0 -9 0 = 9.
■
On the number line, if a first number is to the right of a second number, then the first
number is said to be greater than the second. If the first number is to the left of the second, it is less than the second number. The symbol 7 designates “is greater than,” and the symbol 6 designates “is less than.” These are called signs of inequality. See Fig. 1.4.
E X A M P L E 5 Signs of inequality
Practice Exercises Place the correct sign of inequality ( 6 or 7 ) between the given numbers. 3. - 5 4 4. 0 -3
2 7 -4 2 is to the right of -4
- 4 -2 0
3 6 6 3 is to the
left of 6
246
5 6 9
0 7 -4
-3 7 -7
Pointed toward smaller number
-1 6 0
Fig. 1.4
■
Every number, except zero, has a reciprocal. The reciprocal of a number is 1 divided by the number.
Practice Exercise
5. Find the reciprocals of
(a) - 4
(b)
3 8
E X A M P L E 6 Reciprocal
The
reciprocal
of
7
is
17.
The
reciprocal
of
2 3
is
1
2 3
=
1
*
3 2
=
3 2
invert denominator and multiply (from arithmetic)
The
reciprocal
of
0.5
is
1 0.5
=
2. The reciprocal of
-p is
- p1. Note that the negative sign
# is retained in the reciprocal of a negative number.
We showed the multiplication 11232. We will often find the form
of
1
and
3 2
as
1
*
23.
We
could
with parentheses is preferable.
also
show
it
as
1
3 2
or
■
In applications, numbers that represent a measurement and are written with units of measurement are called denominate numbers. The next example illustrates the use of units and the symbols that represent them.
1.1 Numbers
5
■ For reference, see Appendix B for units of measurement and the symbols used for
them.
E X A M P L E 7 Denominate numbers
(a) To show that a certain TV weighs 62 pounds, we write the weight as 62 lb.
(b) To show that a giant redwood tree is 330 feet high, we write the height as 300 ft.
(c) To show that the speed of a rocket is 1500 meters per second, we write the speed as 1500 m/s. (Note the use of s for second. We use s rather than sec.)
(d) To show that the area of a computer chip is 0.75 square inch, we write the area as 0.75 in.2. (We will not use sq in.)
(e) To show that the volume of water in a glass tube is 25 cubic centimeters, we write
the volume as 25 cm3. (We will not use cu cm nor cc.)
■
It is usually more convenient to state definitions and operations on numbers in a general form. To do this, we represent the numbers by letters, called literal numbers. For example, if we want to say “If a first number is to the right of a second number on the number line, then the first number is greater than the second number,” we can write “If a is to the right of b on the number line, then a 7 b.” Another example of using a literal number is “The reciprocal of n is 1>n.”
Certain literal numbers may take on any allowable value, whereas other literal numbers represent the same value throughout the discussion. Those literal numbers that may vary in a given problem are called variables, and those literal numbers that are held
fixed are called constants.
E X A M P L E 8 variables and constants
(a) The resistance of an electric resistor is R. The current I in the resistor equals the voltage V divided by R, written as I = V>R. For this resistor, I and V may take on various values, and R is fixed. This means I and V are variables and R is a constant. For a different resistor, the value of R may differ.
(b) The fixed cost for a calculator manufacturer to operate a certain plant is b dollars per day, and it costs a dollars to produce each calculator. The total daily cost C to produce n calculators is
C = an + b
Here, C and n are variables, and a and b are constants, and the product of a and n is shown as an. For another plant, the values of a and b would probably differ.
If specific numerical values of a and b are known, say a = $7 per calculator and b = $3000, then C = 7n + 3000. Thus, constants may be numerical or literal. ■
EXERCISES 1.1
In Exercises 1–4, make the given changes in the indicated examples of this section, and then answer the given questions.
1. In the first line of Example 1, change the 5 to -7 and the - 19 to
12. What other changes must then be made in the first paragraph?
2. In Example 4, change the 6 to -6. What other changes must then
be made in the first paragraph?
3. In the left figure of Example 5, change the 2 to - 6. What other
changes must then be made?
4.
In
Example
6,
change
the
2 3
to
32.
What
other
changes
must
then
be
made?
In Exercises 5–8, designate each of the given numbers as being an integer, rational, irrational, real, or imaginary. (More than one designation may be correct.)
5. 3, 2- 4
6.
27 ,
3
-6
7.
-
p, 6
1 8
8. - 2- 6, -2.33
In Exercises 9 and 10, find the absolute value of each real number.
9. 3, - 3,
p, 4
2-1
10. - 0.857,
22,
-
19 ,
4
2-5 -2
6
ChaPTER 1 Basic Algebraic Operations
In Exercises 11–18, insert the correct sign of inequality ( 7 or 6 ) between the given numbers.
11. 6
13. p
15. - 4
17.
-
2 3
8
3.1416
- 0 -30
-
3 4
12. 7 5 14. - 4 0 16. - 22 - 1.42 18. - 0.6 0.2
In Exercises 19 and 20, find the reciprocal of each number.
19. 3, - 4 , y 23 b
20.
-
1 ,
3
0.25,
2x
In Exercises 21 and 22, locate (approximately) each number on a number line as in Fig. 1.2.
21. 2.5,
- 152,
23,
-
3 4
22. - 222,
2p,
11293,
-
7 3
In Exercises 23–46, solve the given problems. Refer to Appendix B for units of measurement and their symbols.
23. Is an absolute value always positive? Explain. 24. Is - 2.17 rational? Explain. 25. What is the reciprocal of the reciprocal of any positive or negative
number?
26. Is the repeating decimal 2.72 rational or irrational?
27. True or False: A nonterminating, nonrepeating decimal is an irrational number.
28. If b 7 a and a 7 0, is 0 b - a 0 6 0 b 0 - 0 a 0 ?
29. List the following numbers in numerical order, starting with the
smallest: -1, 9, p, 25, 0 -8 0 , - 0 -3 0 , -3.1.
30. List the following numbers in numerical order, starting with the
smallest: 15, - 210, - 0 - 6 0 , - 4, 0.25, 0 - p 0 .
31. If a and b are positive integers and b 7 a, what type of number is
represented by the following?
(a) b - a
(b) a - b
(c)
b b
+
a a
32. If a and b represent positive integers, what kind of number is represented by (a) a + b, (b) a>b, and (c) a * b?
33. For any positive or negative integer: (a) Is its absolute value always an integer? (b) Is its reciprocal always a rational number?
34. For any positive or negative rational number: (a) Is its absolute value always a rational number? (b) Is its reciprocal always a rational number?
35. Describe the location of a number x on the number line when (a) x 7 0 and (b) x 6 - 4.
36. Describe the location of a number x on the number line when
(a) 0 x 0 6 1 and (b) 0 x 0 7 2.
37. For a number x 7 1, describe the location on the number line of the reciprocal of x.
38. For a number x 6 0, describe the location on the number line of
the number with a value of 0 x 0 .
39. A complex number is defined as a + bj, where a and b are real numbers and j = 2- 1. For what values of a and b is the complex number a + bj a real number? (All real numbers and all imaginary numbers are also complex numbers.)
40. A sensitive gauge measures the total weight w of a container and the water that forms in it as vapor condenses. It is found that w = c20.1t + 1, where c is the weight of the container and t is the time of condensation. Identify the variables and constants.
41. In an electric circuit, the reciprocal of the total capacitance of two capacitors in series is the sum of the reciprocals of the capacitances a 1 = 1 + 1 b. Find the total capacitance of two capacitances CT C1 C2 of 0.0040 F and 0.0010 F connected in series.
42. Alternating-current (ac) voltages change rapidly between positive and negative values. If a voltage of 100 V changes to - 200 V, which is greater in absolute value?
43. The memory of a certain computer has a bits in each byte. Express the number N of bits in n kilobytes in an equation. (A bit is a single digit, and bits are grouped in bytes in order to represent special characters. Generally, there are 8 bits per byte. If necessary,see Appendix B for the meaning of kilo.)
44. The computer design of the base of a truss is x ft long. Later it is redesigned and shortened by y in. Give an equation for the length L, in inches, of the base in the second design.
45. In a laboratory report, a student wrote “ - 20°C 7 - 30°C.” Is this statement correct? Explain.
46. After 5 s, the pressure on a valve is less than 60 lb/in.2 (pounds per square inch). Using t to represent time and p to represent pressure, this statement can be written “for t 7 5 s, p 6 60 lb/in.2.” In this way, write the statement “when the current I in a circuit is less than 4 A, the resistance R is greater than 12 Ω (ohms).”
answers to Practice Exercises
1. 4.2
2.
-
3 4
3. 6
4. 7
5.
(a)
-
1 4
(b)
8 3
1.2 Fundamental Operations of Algebra
Fundamental Laws of Algebra • operations on Positive and negative Numbers • Order of Operations • operations with Zero
If two numbers are added, it does not matter in which order they are added. (For example, 5 + 3 = 8 and 3 + 5 = 8, or 5 + 3 = 3 + 5.) This statement, generalized and accepted as being correct for all possible combinations of numbers being added, is called
the commutative law for addition. It states that the sum of two numbers is the same,
■ Note carefully the difference: associative law: 5 * 14 * 22 distributive law: 5 * 14 + 22 ■ Note the meaning of identity.
1.2 Fundamental Operations of Algebra
7
regardless of the order in which they are added. We make no attempt to prove this law
in general, but accept that it is true.
In the same way, we have the associative law for addition, which states that the sum
of three or more numbers is the same, regardless of the way in which they are grouped for addition. For example, 3 + 15 + 62 = 13 + 52 + 6.
The laws just stated for addition are also true for multiplication. Therefore, the product
of two numbers is the same, regardless of the order in which they are multiplied, and the
product of three or more numbers is the same, regardless of the way in which they are grouped for multiplication. For example, 2 * 5 = 5 * 2, and 5 * 14 * 22 = 15 * 42 * 2.
Another very important law is the distributive law. It states that the product of
one number and the sum of two or more other numbers is equal to the sum of the
products of the first number and each of the other numbers of the sum. For
example,
514 + 22 = 5 * 4 + 5 * 2
In this case, it can be seen that the total is 30 on each side.
In practice, we use these fundamental laws of algebra naturally without thinking
about them, except perhaps for the distributive law.
Not all operations are commutative and associative. For example, division is not com-
mutative,
because
the
order
of
division
of
two
numbers
does
matter.
For
instance,
6 5
∙
5 6
( ∙ is read “does not equal”). (Also, see Exercise 54.)
Using literal numbers, the fundamental laws of algebra are as follows:
Commutative law of addition: a ∙ b ∙ b ∙ a Associative law of addition: a ∙ 1 b ∙ c2 ∙ 1 a ∙ b2 ∙ c Commutative law of multiplication: ab ∙ ba Associative law of multiplication: a1 bc2 ∙ 1 ab2 c Distributive law: a1 b ∙ c2 ∙ ab ∙ ac
Each of these laws is an example of an identity, in that the expression to the left of the = sign equals the expression to the right for any value of each of a, b, and c.
OPERATIONS ON POSITIVE AND NEGATIVE NUMBERS When using the basic operations (addition, subtraction, multiplication, division) on positive and negative numbers, we determine the result to be either positive or negative according to the following rules.
Addition of two numbers of the same sign Add their absolute values and assign the sum their common sign.
■ From Section 1.1, we recall that a positive number is preceded by no sign. Therefore, in using these rules, we show the “sign” of a positive number by simply writing the number itself.
E X A M P L E 1 adding numbers of the same sign
(a) 2 + 6 = 8
the sum of two positive numbers is positive
(b) -2 + 1 -62 = - 12 + 62 = -8
the sum of two negative numbers is negative
The negative number -6 is placed in parentheses because it is also preceded
by a plus sign showing addition. It is not necessary to place the -2 in
parentheses.
■
8
ChaPTER 1 Basic Algebraic Operations
Addition of two numbers of different signs Subtract the number of smaller absolute value from the number of larger absolute value and assign to the result the sign of the number of larger absolute value.
E X A M P L E 2 adding numbers of different signs
(a) 2 + 1 -62 = - 16 - 22 = -4
(b) -6 + 2 = - 16 - 22 = -4
(c)
6 + 1 -22 = 6 - 2 = 4
(d)
-2 + 6 = 6 - 2 = 4
the negative 6 has the larger absolute value the positive 6 has the larger absolute value
the subtraction of absolute values
■
Subtraction of one number from another Change the sign of the number being subtracted and change the subtraction to addition. Perform the addition.
Practice Exercises Evaluate: 1. - 5 - 1 -82 2. - 51 - 82
noTE →
E X A M P L E 3 subtracting positive and negative numbers (a) 2 - 6 = 2 + 1 -62 = - 16 - 22 = -4
Note that after changing the subtraction to addition, and changing the sign of 6 to make it -6, we have precisely the same illustration as Example 2(a).
(b) -2 - 6 = -2 + 1 -62 = - 12 + 62 = -8
Note that after changing the subtraction to addition, and changing the sign of 6 to make it -6, we have precisely the same illustration as Example 1(b).
(c) -a - 1 -a2 = -a + a = 0
This shows that subtracting a number from itself results in zero, even if the number
is negative. [Subtracting a negative number is equivalent to adding a positive number of the same absolute value.]
(d) -2 - 1 -62 = -2 + 6 = 4
(e) The change in temperature from -12°C to -26°C is
-26°C - 1 -12°C2 = -26°C + 12°C = -14°C
■
Multiplication and division of two numbers The product (or quotient) of two numbers of the same sign is positive. The product (or quotient) of two numbers of different signs is negative.
E X A M P L E 4 multiplying and dividing positive and negative numbers
(a) 31122 = 3 * 12 = 36 (b) -31 -122 = 3 * 12 = 36 (c) 31 -122 = - 13 * 122 = -36 (d) -31122 = - 13 * 122 = -36
12 =4
3
- 12 -3
=
4
- 12 3
=
-
12 3
=
-4
12 -3
=
-
12 3
=
-4
result is positive if both numbers are positive
result is positive if both numbers are negative
result is negative if one number is positive and the other is negative
■
ORDER OF OPERATIONS
Often, how we are to combine numbers is clear by grouping the numbers using symbols such as parentheses, ( ); the bar, ____, between the numerator and denominator of a fraction; and vertical lines for absolute value. Otherwise, for an expression in which there are several operations, we use the following order of operations.
1.2 Fundamental Operations of Algebra
9
order of operations 1. Perform operations within grouping symbols (parentheses, brackets, or abso-
lute value symbols). 2. Perform multiplications and divisions (from left to right). 3. Perform additions and subtractions (from left to right).
■
Note that 20
,
12
+
32
=
2
20 +
3,
whereas 20
,
2
+
3
=
20 2
+
3.
noTE →
Practice Exercises Evaluate: 3. 12 - 6 , 2 4. 16 , 12 * 42
E X A M P L E 5 order of operations
(a) 20 , 12 + 32 is evaluated by first adding 2 + 3 and then dividing. The grouping of 2 + 3 is clearly shown by the parentheses. Therefore, 20 , 12 + 32 = 20 , 5 = 4.
(b) 20 , 2 + 3 is evaluated by first dividing 20 by 2 and then adding. No specific grouping is shown, and therefore the division is done before the addition. This means 20 , 2 + 3 = 10 + 3 = 13.
(c) [16 - 2 * 3 is evaluated by first multiplying 2 by 3 and then subtracting. We do not first subtract 2 from 16.] Therefore, 16 - 2 * 3 = 16 - 6 = 10.
(d) 16 , 2 * 4 is evaluated by first dividing 16 by 2 and then multiplying. From left to right, the division occurs first. Therefore, 16 , 2 * 4 = 8 * 4 = 32.
(e) 0 3 - 5 0 - 0 -3 - 6 0 is evaluated by first performing the subtractions within the
absolute value vertical bars, then evaluating the absolute values, and then subtract-
ing. This means that 0 3 - 5 0 - 0 -3 - 6 0 = 0 -2 0 - 0 -9 0 = 2 - 9 = -7. ■
When evaluating expressions, it is generally more convenient to change the operations and numbers so that the result is found by the addition and subtraction of positive numbers. When this is done, we must remember that
a + 1 -b2 = a - b
(1.1)
a - 1 -b2 = a + b
(1.2)
Practice Exercises Evaluate: 5. 21 -32
-
4
2
8
6.
0 5 - 15 0
2
-
-9 3
3000 lb 40 mi/h
2000 lb 20 mi/h
16 mi/h Fig. 1.5
E X A M P L E 6 Evaluating numerical expressions
(a) 7 + 1 -32 - 6 = 7 - 3 - 6 = 4 - 6 = -2
using Eq. (1.1)
(b)
18 -6
+5-
1 -22132
=
-3 + 5 -
1 -62
=2+6=8
using Eq. (1.2)
(c)
0 3 - 15 0
-2
-
4
8 -
6
=
12 -2
-
8 -2
=
-6
-
1 -42
=
-6
+
4
=
-2
(d)
- 12 2-8
+
5-1 21 -12
=
- 12 -6
+
4 -2
=
2
+
1 -22
=
2
-
2
=
0
In illustration (b), we see that the division and multiplication were done before the addi-
tion and subtraction. In (c) and (d), we see that the groupings were evaluated first. Then
we did the divisions, and finally the subtraction and addition.
■
E X A M P L E 7 Evaluating—velocity after collision
A 3000-lb van going at 40 mi/h ran head-on into a 2000-lb car going at 20 mi/h. An
insurance investigator determined the velocity of the vehicles immediately after the col-
lision from the following calculation. See Fig. 1.5.
30001402 + 1200021 -202 120,000 + 1 -40,0002 120,000 - 40,000
3000 + 2000
=
3000 + 2000
=
5000
=
80,000 5000
=
16 mi/h
The numerator and the denominator must be evaluated before the division is performed.
The multiplications in the numerator are performed first, followed by the addition in the
denominator and the subtraction in the numerator.
■
10
ChaPTER 1 Basic Algebraic Operations
OPERATIONS wITH ZERO
Because operations with zero tend to cause some difficulty, we will show them here. If a is a real number, the operations of addition, subtraction, multiplication, and divi-
sion with zero are as follows:
a+0=a a-0=a a*0=0
0 - a = -a
0
,
a
=
0 a
=
0
1if a ∙ 02
∙ means “is not equal to”
E X A M P L E 8 operations with zero
(a) 5 + 0 = 5
(b) -6 - 0 = -6
0 (d) = 0
6
0 (e) -3 = 0
(c) 0 - 4 = -4
(f)
5*0 7
=
0 7
=0
■
Note that there is no result defined for division by zero. To understand the reason for
this,
consider
the
results
for
6 2
and
60.
6 =3
2
since
2*3=6
If
6 0
=
b, then
0
*
b
=
6. This cannot be true because
0
*
b
=
0
for any value of b.
Thus, division by zero is undefined.
(The
special
case
of
0 0
is
termed
indeterminate.
If
0 0
=
b,
then
0
=
0
*
b,
which
is
true for any value of b. Therefore, no specific value of b can be determined.)
E X A M P L E 9 Division by zero is undefined
2 5
,
0 is undefined
8 is undefined
0
7 0
* *
0 6
is indeterminate
■
see above
CAUTION The operations with zero will not cause any difficulty if we remember to never divide by zero. ■
Division by zero is the only undefined basic operation. All the other operations with zero may be performed as for any other number.
EXERCISES 1.2
In Exercises 1–4, make the given changes in the indicated examples of this section, and then solve the resulting problems.
1. In Example 5(c), change 3 to 1 - 22 and then evaluate. 2. In Example 6(b), change 18 to - 18 and then evaluate. 3. In Example 6(d), interchange the 2 and 8 in the first denominator
and then evaluate. 4. In the rightmost illustration in Example 9, interchange the 6 and
the 0 above the 6. Is any other change needed?
In Exercises 5–38, evaluate each of the given expressions by performing the indicated operations.
5. 5 + 1 - 82
6. - 4 + 1 -72
7. - 3 + 9
8. 18 - 21
11. 71 - 42
14.
-9 3
9. - 19 - 1 -162
12. - 9132
-6120 - 102
15.
-3
10. - 8 - 1 -102
13. - 71 - 52
16.
- 28 -715 - 62
17. - 21421 -52 - 64
20. -2 0 4 - 8 0 23. -9 - 0 2 - 10 0
17 - 72122 26. 17 - 721 -12
18. - 31 - 421 - 62 19. 212 - 72 , 10
21. 16 , 21 -42
22. - 20 , 51 - 42
24.
17
-
72
,
15
-
72 25.
17 - 7 7-7
27. 8 - 31 - 42
28. - 20 + 8 , 4
29.
-21 -62
+
`
8 -2
`
30.
0 -20
-2
- 1 -221 -52
31. 101 - 821 -32 , 110 - 502
7 - 0 -50
32. - 11 - 22
33.
3
24 + 1 -52
-
41 -92
,
1 -32
34.
- 18 3
-
4 - 0 -60
-1
35.
-7
-
0 -14 0
212 - 32
-
306
-
80
36.
-71 -32
+
-6 -3
-
0 -90
3 0 -9 - 21-32 0
37. 1 + 1 - 102
201 - 122 - 401 - 152
38.
98 - 0 -98 0
In Exercises 39–46, determine which of the fundamental laws of algebra is demonstrated.
39. 6172 = 7162 41. 613 + 12 = 6132 + 6112
40. 6 + 8 = 8 + 6 42. 415 * p2 = 14 * 521p2
43. 3 + 15 + 92 = 13 + 52 + 9 44. 813 - 22 = 8132 - 8122
45. 1 25 * 32 * 9 = 25 * 13 * 92 46. 13 * 62 * 7 = 7 * 13 * 62
In Exercises 47–50, for numbers a and b, determine which of the following expressions equals the given expression.
(a) a + b 47. - a + 1 -b2 49. - b - 1 -a2
(b) a - b
(c) b - a 48. b - 1 - a2 50. - a - 1 -b2
(d) - a - b
In Exercises 51–66, solve the given problems. Refer to Appendix B for units of measurement and their symbols.
51. Insert the proper sign 1 =, 7, 6 2 to make the following true:
0 5 - 1-22 0 0 -5 - 0 -2 0 0
52. Insert the proper sign 1 =, 7, 6 2 to make the following true:
0 -3 - 0 - 70 0 0 0 -30 - 70
53. (a) What is the sign of the product of an even number of negative numbers? (b) What is the sign of the product of an odd number of negative numbers?
54. Is subtraction commutative? Explain.
55. Explain why the following definition of the absolute value of a real number x is either correct or incorrect (the symbol Ú means “is
equal to or greater than”): If x Ú 0, then 0 x 0 = x; if x 6 0, then 0 x 0 = -x.
1.2 Fundamental Operations of Algebra
11
56. Explain what is the error if the expression 24 - 6 , 2 # 3 is evalu-
ated as 27. What is the correct value?
57. Desxcr-ibey the values of x and y for which (a) - xy = 1 and (b) x - y = 1.
58. Describe the values of x and y for which (a) 0 x + y 0 = 0 x 0 + 0 y 0 and (b) 0 x - y 0 = 0 x 0 + 0 y 0 .
59. The changes in the price of a stock (in dollars) for a given week were - 0.68, + 0.42, + 0.06, - 0.11, and + 0.02. What was the total change in the stock’s price that week?
60. Using subtraction of signed numbers, find the difference in the altitude of the bottom of the Dead Sea, 1396 m below sea level, and the bottom of Death Valley, 86 m below sea level.
61. Some solar energy systems are used to supplement the utility com-
pany power supplied to a home such that the meter runs backward
if the solar energy being generated is greater than the energy being
used. 3.0-h
With such a system, period and only 2.1
ikfWthe#
solar power averages h is used during this
1.5 kW period,
for a what
will be the change in the meter reading for this period? Hint: Solar
power generated makes the meter run in the negative direction
while power used makes it run in the positive direction.
62. A baseball player’s batting average (total number of hits divided by total number of at-bats) is expressed in decimal form from 0.000 (no hits for all at-bats) to 1.000 (one hit for each at-bat). A player’s batting average is often shown as 0.000 before the first at-bat of the season. Is this a correct batting average? Explain.
63. The daily high temperatures (in °C) for Edmonton, Alberta, in the first week in March were recorded as - 7, - 3, 2, 3, 1, - 4, and - 6. What was the average daily temperature for the week? (Divide the
algebraic sum of readings by the number of readings.)
64. A flare is shot up from the top of a tower. Distances above the flare
gun are positive and those below it are negative. After 5 s the verti-
cal distance (in ft) of the flare from the flare gun is found by evaluating 1702152 + 1 - 1621252. Find this distance.
65. Find the sum of the voltages of the batteries shown in Fig. 1.6. Note the directions in which they are connected.
+6 V
-+ -2 V
+8 V
Fig. 1.6
-+ -5 V
+3 V
66. A faulty gauge on a fire engine pump caused the apparent pressure in the hose to change every few seconds. The pressures (in lb/in.2 above and below the set pressure were recorded as: + 7, -2, - 9, -6. What was the change between (a) the first two readings, (b) between the middle two readings, and (c) the last two readings?
67. One oil-well drilling rig drills 100 m deep the first day and 200 m deeper the second day. A second rig drills 200 m deep the first day and 100 m deeper the second day. In showing that the total depth drilled by each rig was the same, state what fundamental law of algebra is illustrated.
68. A water tank leaks 12 gal each hour for 7 h, and a second tank leaks 7 gal each hour for 12 h. In showing that the total amount leaked is the same for the two tanks, what fundamental law of algebra is illustrated?
12
ChaPTER 1 Basic Algebraic Operations
69. On each of the 7 days of the week, a person spends 25 min on Facebook and 15 min on Twitter. Set up the expression for the total time spent on these two sites that week. What fundamental law of algebra is illustrated?
70. A jet travels 600 mi/h relative to the air. The wind is blowing at 50 mi/h. If the jet travels with the wind for 3 h, set up the
expression for the distance traveled. What fundamental law of algebra is illustrated?
answers to Practice Exercises 1. 3 2. 40 3. 9 4. 2 5. - 4 6. 8
1.3 Calculators and Approximate Numbers
Graphing Calculators • Approximate Numbers • Significant Digits • Accuracy and Precision • Rounding Off • Operations with Approximate Numbers • Estimating Results
■ The calculator screens shown with text material are for a TI-84 Plus. They are intended only as an illustration of a calculator screen for the particular operation. Screens for other models may differ.
You will be doing many of your calculations on a calculator, and a graphing calculator can be used for these calculations and many other operations. In this text, we will restrict our coverage of calculator use to graphing calculators because a scientific calculator cannot perform many of the required operations we will cover.
A brief discussion of the graphing calculator appears in Appendix C, and sample calculator screens appear throughout the book. Since there are many models of graphing calculators, the notation and screen appearance for many operations will differ from one model to another. You should practice using your calculator and review its manual to be sure how it is used. Following is an example of a basic calculation done on a graphing calculator.
Fig. 1.7
■ When less than half of a calculator screen is needed, a partial screen will be shown.
E X A M P L E 1 Calculating on a graphing calculator
In order to calculate the value of 38.3 - 12.91 -3.582, the numbers are entered as follows. The calculator will perform the multiplication first, following the order of operations shown in Section 1.2. The sign of -3.58 is entered using the 1 - 2 key, before 3.58 is entered. The display on the calculator screen is shown in Fig. 1.7.
38.3 - 12.9 * 1 - 2 3.58 ENTER keystrokes
This means that 38.3 - 12.91 -3.582 = 84.482. Note in the display that the negative sign of -3.58 is smaller and a little higher to
distinguish it from the minus sign for subtraction. Also note the * shown for multiplica-
tion; the asterisk is the standard computer symbol for multiplication.
■
■ Some calculator keys on different models are labeled differently. For example, on some models, the EXE key is equivalent to the ENTER key.
■ Calculator keystrokes for various operations can be found by using the URLs given in this text. A list of all the calculator instructions is at goo.gl/eAUgW3.
Looking back into Section 1.2, we see that the minus sign is used in two different ways: (1) to indicate subtraction and (2) to designate a negative number. This is clearly shown on a graphing calculator because there is a key for each purpose. The - key is used for subtraction, and the 1 - 2 key is used before a number to make it negative.
We will first use a graphing calculator for the purpose of graphing in Section 3.5. Before then, we will show some calculational uses of a graphing calculator.
APPROXIMATE NUMBERS AND SIGNIFICANT DIGITS Most numbers in technical and scientific work are approximate numbers, having been determined by some measurement. Certain other numbers are exact numbers, having been determined by a definition or counting process.
E X A M P L E 2 approximate numbers and exact numbers
One person measures the distance between two cities on a map as 36 cm, and another
person measures it as 35.7 cm. However, the distance cannot be measured exactly.
If a computer prints out the number of names on a list of 97, this 97 is exact. We know
it is not 96 or 98. Since 97 was found from precise counting, it is exact.
By definition, 60 s = 1 min, and the 60 and the 1 are exact.
■
1.3 Calculators and Approximate Numbers
13
An approximate number may have to include some zeros to properly locate the decimal point. Except for these zeros, all other digits are called significant digits.
■ To show that zeros at the end of a whole
number are significant, a notation that can
be used is significant
tzoeprola. cUesiangbathriosvneorttahteiolna,s7t8,00–0
is shown to have four significant digits.
Practice Exercises Determine the number of significant digits. 1. 1010 2. 0.1010
E X A M P L E 3 significant digits
All numbers in this example are assumed to be approximate. (a) 34.7 has three significant digits.
(b) 0.039 has two significant digits. The zeros properly locate the decimal point.
(c) 706.1 has four significant digits. The zero is not used for the location of the decimal point. It shows the number of tens in 706.1.
(d) 5.90 has three significant digits. The zero is not necessary as a placeholder and should not be written unless it is significant.
(e) 1400 has two significant digits, unless information is known about the number that makes either or both zeros significant. Without such information, we assume that the zeros are placeholders for proper location of the decimal point.
(f) Other approximate numbers with the number of significant digits are 0.0005 (one),
960,000 (two), 0.0709 (three), 1.070 (four), and 700.00 (five).
■
From Example 3, we see that all nonzero digits are significant. Also, zeros not used as placeholders (for location of the decimal point) are significant.
In calculations with approximate numbers, the number of significant digits and the position of the decimal point are important. The accuracy of a number refers to the number of significant digits it has, whereas the precision of a number refers to the decimal position of the last significant digit.
E X A M P L E 4 accuracy and precision
One technician measured the thickness of a metal sheet as 3.1 cm and another technician
measured it as 3.12 cm. Here, 3.12 is more precise since its last digit represents hun-
dredths and 3.1 is expressed only to tenths. Also, 3.12 is more accurate since it has three
significant digits and 3.1 has only two.
A concrete driveway is 230 ft long and 0.4 ft thick. Here, 230 is more accurate (two
significant digits) and 0.4 is more precise (expressed to tenths).
■
The last significant digit of an approximate number is not exact. It has usually been determined by estimating or rounding off. However, it is not off by more than one-half of a unit in its place value.
E X A M P L E 5 meaning of the last digit of an approximate number
When we write the measured distance on the map in Example 2 as 35.7 cm, we are saying
that the distance is at least 35.65 cm and no more than 35.75 cm. Any value between
these, rounded off to tenths, would be 35.7 cm.
In
changing
the
fraction
2 3
to
the
approximate
decimal
value
0.667,
we
are
saying
that
the value is between 0.6665 and 0.6675.
■
■ On graphing calculators, it is possible to set the number of decimal places (to the right of the decimal point) to which results will be rounded off.
To round off a number to a specified number of significant digits, discard all digits to the right of the last significant digit (replace them with zeros if needed to properly place the decimal point). If the first digit discarded is 5 or more, increase the last significant digit by 1 (round up). If the first digit discarded is less than 5, do not change the last significant digit (round down).
14
ChaPTER 1 Basic Algebraic Operations
E X A M P L E 6 Rounding off
Practice Exercises
noTE →
Round off each number to three significant
digits.
3. 2015 4. 0.3004
(a) 70,360 rounded off to three significant digits is 70,400. Here, 3 is the third significant digit and the next digit is 6. Because 6 7 5, the 3 is rounded up to 4 and the 6 is replaced with a zero to hold the place value.
(b) 70,430 rounded off to three significant digits, or to the nearest hundred, is 70,400. Here the 3 is replaced with a zero.
(c) 187.35 rounded off to four significant digits, or to tenths, is 187.4.
(d) 187.349 rounded off to four significant digits is 187.3. We do not round up the 4 and then round up the 3.
(e) 35.003 rounded off to four significant digits is 35.00. [We do not discard the zeros because they are significant and are not used only to properly place the decimal point.]
(f) 849,720 rounded off to three significant digits is 850,000. The bar over the zero
shows that digit is significant.
■
noTE →
OPERATIONS wITH APPROXIMATE NUMBERS
[When performing operations on approximate numbers, we must express the result to an accuracy or precision that is valid.] Consider the following examples.
16.3 ft 0.927 ft
17.227 ft
E X A M P L E 7 Precision—length of pipe
smallest values largest values
16.25 ft 0.9265 ft
16.35 ft 0.9275 ft
17.1765 ft
17.2775 ft
A pipe is made in two sections. One is measured as 16.3 ft long and the other as 0.927 ft
long. What is the total length of the two sections together?
It may appear that we simply add the numbers as shown at the left. However, both
numbers are approximate, and adding the smallest possible values and the largest pos-
sible values, the result differs by 0.1 (17.2 and 17.3) when rounded off to tenths. Rounded
off to hundredths (17.18 and 17.28), they do not agree at all because the tenths digit is
different. Thus, we get a good approximation for the total length if it is rounded off to
tenths, the precision of the least precise length, and it is written as 17.2 ft.
■
0.005 ft
207.54 ft 16,900 ft2 Fig. 1.8
0.05 ft 81.4 ft
E X A M P L E 8 accuracy—area of land plot
We find the area of the rectangular piece of land in Fig. 1.8 by multiplying the
length, 207.54 ft, by the width, 81.4 ft. Using a calculator, we find that 1207.542 181.42 = 16,893.756. This apparently means the area is 16,893.756 ft2.
However, the area should not be expressed with this accuracy. Because the length
and width are both approximate, we have
1207.535 ft2 181.35 ft2 = 16,882.97225 ft2 1207.545 ft2 181.45 ft2 = 16,904.54025 ft2
least possible area greatest possible area
These values agree when rounded off to three significant digits 116,900 ft22 but do not
agree when rounded off to a greater accuracy. Thus, we conclude that the result is accu-
rate only to three significant digits, the accuracy of the least accurate measurement, and
that the area is written as 16,900 ft2.
■
■ The results of operations on approximate numbers shown at the right are based on reasoning that is similar to that shown in Examples 7 and 8.
The Result of operations on approximate numbers
1. When approximate numbers are added or subtracted, the result is expressed with the precision of the least precise number.
2. When approximate numbers are multiplied or divided, the result is expressed with the accuracy of the least accurate number.
3. When the root of an approximate number is found, the result is expressed with the accuracy of the number.
4. When approximate numbers and exact numbers are involved, the accuracy of the result is limited only by the approximate numbers.
1.3 Calculators and Approximate Numbers
15
CAUTION Always express the result of a calculation with the proper accuracy or precision. When using a calculator, if additional digits are displayed, round off the final result (do not round off in any of the intermediate steps). ■
■ When rounding off a number, it may seem difficult to discard the extra digits. However, if you keep those digits, you show a number with too great an accuracy, and it is incorrect to do so.
E X A M P L E 9 adding approximate numbers
Find the sum of the approximate numbers 73.2, 8.0627, and 93.57. Showing the addition in the standard way and using a calculator, we have
73.2 8.0627 93.57 174.8327
least precise number (expressed to tenths) final display must be rounded to tenths
Therefore, the sum of these approximate numbers is 174.8.
■
Practice Exercise Evaluate using a calculator.
5.
40.5
+
3275 - 60.041
1Numbers are approximate.2
E X A M P L E 1 0 multiplying approximate numbers
In finding the product of the approximate numbers 2.4832 and 30.5 on a calculator, the
final display shows 75.7376. However, since 30.5 has only three significant digits, the
product is 75.7.
■
E X A M P L E 1 1 Combined operations
For problems with multiple operations, follow the correct order of operations as given in Section 1.2. Keep all the digits in the intermediate steps, but keep track of (perhaps by underlining) the significant digits that would be retained according to the appropriate rounding rule for each step. Then round off the final answer according to the last operation that is performed. For example,
14.265 * 2.602 , 13.7 + 5.142 = 11.089 , 8.84 = 1.3
Note that three significant digits are retained from the multiplication and one decimal
place precision is retained from the addition. The final answer is rounded off to two
significant digits, which is the accuracy of the least accurate number in the final division
(based on the underlined significant digits).
■
E X A M P L E 1 2 operations with exact numbers and approximate numbers
Using the exact number 600 and the approximate number 2.7, we express the result to tenths if the numbers are added or subtracted. If they are multiplied or divided, we express the result to two significant digits. Since 600 is exact, the accuracy of the result depends only on the approximate number 2.7.
600 + 2.7 = 602.7 600 * 2.7 = 1600
600 - 2.7 = 597.3 600 , 2.7 = 220
There are 16 pieces in a pile of lumber and the average length of a piece is 482 mm.
Here 16 is exact, but 482 is approximate. To get the total length of the pieces in the pile, the product 16 * 482 = 7712 must be rounded off to three significant digits, the accu-
racy of 482. Therefore, we can state that the total length is about 7710 mm.
■
noTE →
[A note regarding the equal sign 1 = 2 is in order. We will use it for its defined mean-
ing of “equals exactly” and when the result is an approximate number that has been
properly rounded off.] Although 227.8 ≈ 5.27, where ≈ means “equals approximately,”
we write 227.8 = 5.27, since 5.27 has been properly rounded off. You should make a rough estimate of the result when using a calculator. An estimation
may prevent accepting an incorrect result after using an incorrect calculator sequence, particularly if the calculator result is far from the estimated value.
16
ChaPTER 1 Basic Algebraic Operations
E X A M P L E 1 3 Estimating results
In Example 1, we found that 38.3 - 12.91 -3.582 = 84.482
using exact numbers
When using the calculator, if we forgot to make 3.58 negative, the display would be -7.882, or if we incorrectly entered 38.3 as 83.3, the display would be 129.482.
However, if we estimate the result as
40 - 101 -42 = 80
we know that a result of -7.882 or 129.482 cannot be correct.
When estimating, we can often use one-significant-digit approximations. If the cal-
culator result is far from the estimate, we should do the calculation again.
■
EXERCISES 1.3
In Exercises 1–4, make the given changes in the indicated examples of this section, and then solve the given problems.
1. In Example 3(b), change 0.039 to 0.390. Is there any change in the conclusion?
2. In Example 6(e), change 35.003 to 35.303 and then find the result.
3. In Example 10, change 2.4832 to 2.5 and then find the result. 4. In Example 13, change 12.9 to 21.9 and then find the estimated
value.
In Exercises 5–10, determine whether the given numbers are approximate or exact.
5. A car with 8 cylinders travels at 55 mi/h. 6. A computer chip 0.002 mm thick is priced at $7.50. 7. In 24 h there are 1440 min. 8. A calculator has 50 keys, and its battery lasted for 50 h of use. 9. A cube of copper 1 cm on an edge has a mass of 9 g. 10. Of a building’s 90 windows, 75 were replaced 15 years ago.
In Exercises 11–18, determine the number of significant digits in each of the given approximate numbers.
11. 107; 3004; 1040 13. 6.80; 6.08; 0.068 15. 3000; 3000.1; 3000.10 17. 5000; 5000.0; 5000
12. 3600; 730; 2055 14. 0.8730; 0.0075; 0.0305 16. 1.00; 0.01; 0.0100 18. 200; 200; 200.00
In Exercises 19–24, determine which of the pair of approximate numbers is (a) more precise and (b) more accurate.
19. 30.8; 0.010
20. 0.041; 7.673
21. 0.1; 78.0 23. 7000; 0.004
22. 7040; 0.004
24. 50.060; 0 -8.914 0
In Exercises 25–32, round off the given approximate numbers (a) to three significant digits and (b) to two significant digits.
25. 4.936 29. 5968
26. 80.53 30. 30.96
27. - 50.893 31. 0.9449
28. 7.004 32. 0.9999
In Exercises 33–42, perform the indicated operations assuming all numbers are approximate. Round your answers using the procedure shown in Example 11.
33. 12.78 + 1.0495 - 1.633 35. 0.6572 * 3.94 - 8.651
37. 8.75 + 11.2213.842
8.75115.322 39. 8.75 + 15.32
41.
4.52
-
2.0561309.62 395.2
34. 3.64(17.06)
36. 41.5 - 26.4 , 3.7
38.
28
-
20.955 2.2
8.9714.0032 40. 2.0 + 4.78
42.
8.195
+
14.9 1.7 + 2.1
In Exercises 43–46, perform the indicated operations. The first number is approximate, and the second number is exact.
43. 0.9788 + 14.9 45. - 3.1421652
44. 17.311 - 22.98 46. 8.62 , 1728
In Exercises 47–50, answer the given questions. Refer to Appendix B for units of measurement and their symbols.
47. The manual for a heart monitor lists the frequency of the ultrasound wave as 2.75 MHz. What are the least possible and the greatest possible frequencies?
48. A car manufacturer states that the engine displacement for a certain model is 2400 cm3. What should be the least possible and greatest possible displacements?
49. A flash of lightning struck a tower 3.25 mi from a person. The thunder was heard 15 s later. The person calculated the speed of sound and reported it as 1144 ft/s. What is wrong with this conclusion?
50. A technician records 4.4 s as the time for a robot arm to swing from the extreme left to the extreme right, 2.72 s as the time for the return swing, and 1.68 s as the difference in these times. What is wrong with this conclusion?
In Exercises 51–58, perform the calculations on a calculator without rounding.
51. Evaluate: (a) 2.2 + 3.8 * 4.5 (b) 12.2 + 3.82 * 4.5 52. Evaluate: (a) 6.03 , 2.25 + 1.77 (b) 6.03 , 12.25 + 1.772
1.4 Exponents and Unit Conversions
17
53. Evaluate: (a) 2 + 0 (b) 2 - 0 (c) 0 - 2 (d) 2 * 0 (e) 2 , 0 Compare with operations with zero on page 10.
54. Evaluate: (a) 2 , 0.0001 and 2 , 0 (b) 0.0001 , 0.0001 and 0 , 0 (c) Explain why the displays differ.
55. Show that p is not equal exactly to (a) 3.1416, or (b) 22>7.
56. At some point in the decimal equivalent of a rational number, some sequence of digits will start repeating endlessly. An irrational number never has an endlessly repeating sequence of digits. Find the decimal equivalents of (a) 8>33 and (b) p. Note the repetition for 8>33 and that no such repetition occurs for p.
57. Following Exercise 56, show that the decimal equivalents of the following fractions indicate they are rational: (a) 1>3 (b) 5>11 (c) 2>5. What is the repeating part of the decimal in (c)?
58. Following Exercise 56, show that the decimal equivalent of the fraction 124>990 indicates that it is rational. Why is the last digit different?
In Exercises 59–64, assume that all numbers are approximate unless stated otherwise.
59. In 3 successive days, a home solar system produced 32.4 MJ, 26.704 MJ, and 36.23 MJ of energy. What was the total energy produced in these 3 days?
60. A shipment contains eight plasma televisions, each weighing 68.6 lb, and five video game consoles, each weighing 15.3 lb. What is the total weight of the shipment?
61. Certain types of iPhones and iPads weigh approximately 129 g and 298.8 g, respectively. What is the total weight of 12 iPhones and 16 iPads of these types? (Source: Apple.com.)
62. Find the voltage in a certain electric circuit by multiplying the sum of the resistances 15.2 Ω, 5.64 Ω, and 101.23 Ω by the current 3.55 A.
63. The percent of alcohol in a certain car engine coolant is found by 100140.63 + 52.962
performing the calculation 105.30 + 52.96 . Find this percent of alcohol. The number 100 is exact.
64. The tension (in N) in a cable lifting a crate at a construction site
50.4519.802
was found by calculating the value of 1
+
100.9
,
, where the 23
1 is exact. Calculate the tension.
In Exercises 65 and 66, all numbers are approximate. (a) Estimate the result mentally using one-significant-digit approximations of all the numbers, and (b) compute the result using the appropriate rounding rules and compare with the estimate.
65. 7.84 * 4.932 - 11.317
66. 21.6 - 53.14 , 9.64
answers to Practice Exercises 1. 3 2. 4 3. 2020 4. 0.300 5. - 14.0
1.4 Exponents and Unit Conversions
Positive Integer Exponents • Zero and Negative Exponents • Order of Operations • Evaluating Algebraic Expressions • Converting units
In mathematics and its applications, we often have a number multiplied by itself several times. To show this type of product, we use the notation an, where a is the number and n is the number of times it appears. In the expression an, the number a is called the base, and n is called the exponent; in words, an is read as “the nth power of a.”
E X A M P L E 1 meaning of exponents
■ Two forms are shown for Eqs. (1.4) in order that the resulting exponent is a positive integer. We consider negative and zero exponents after the next three examples.
(a) 4 * 4 * 4 * 4 * 4 = 45
the fifth power of 4
(b) 1 - 22 1 - 22 1 - 22 1 - 22 = 1 - 224 the fourth power of -2
(c) a * a = a2
the second power of a, called “a squared”
111
13
(d) a b a b a b = a b
555
5
the third power of 15, called “15 cubed”
■
We now state the basic operations with exponents using positive integers as exponents. Therefore, with m and n as positive integers, we have the following operations:
am * an = am + n
am an
=
am - n1m
7
n, a
∙
02
1am2n = amn
1ab2n = anbn
aabn b
=
an bn
am
1
an = an - m
1b ∙ 02
1m 6 n, a ∙ 02
(1.3) (1.4) (1.5) (1.6)
18
ChaPTER 1 Basic Algebraic Operations
E X A M P L E 2 Illustrating Eqs. (1.3) and (1.4)
■ In a3, which equals a * a * a, each a is called a factor. A more general definition of factor is given in Section 1.7.
■ Here we are using the fact that a (not zero) divided by itself equals 1, or a>a = 1.
■ Note that Eq. (1.3) can be verified numerically, for example, by 23 * 25 = 8 * 32 = 256 23 * 25 = 23 + 5 = 28 = 256
Using Eq. (1.3):
Using the meaning of exponents:
add exponents
a3 * a5 = a3 + 5 = a8
8 factors of a (3 factors of a)(5 factors of a)
a3 * a5 = 1a * a * a21a * a * a * a * a2 = a8
Using first form Eq. (1.4): Using the meaning of exponents:
573
a5 a3
=
a5 - 3
=
a2
a5 a3
=
1 a
*
1 a a
* *
1 a a
* *
a a
*
a
=
a2
111
Using second form Eq. (1.4): Using the meaning of exponents:
a3
1
1
a5 = a5 - 3 = a2
a3
1 a
*
1 a
*
1 a
1
a5 = a * a * a * a * a = a2
573
111
■
E X A M P L E 3 Illustrating Eqs. (1.5) and (1.6)
Using Eq. (1.5):
multiply exponents
1a523 = a5132 = a15 Using first form Eq. (1.6):
Using the meaning of exponents:
1a523 = 1a52 1a52 1a52 = a5 + 5 + 5 = a15 Using the meaning of exponents:
1ab23 = a3b3 Using second form Eq. (1.6):
1ab23 = 1ab2 1ab2 1ab2 = a3b3 Using the meaning of exponents:
a 3 a3
ab b
= b3
a3
aaa
a3
ab b
=
a ba ba b bbb
=
b3
■
CAUTION When an expression involves a product or a quotient of different bases, only exponents of the same base may be combined. ■ Consider the following example.
E X A M P L E 4 other illustrations of exponents
CAUTION In illustration (b), note that ax2
means a times the square of x and does not mean a2x2, whereas 1ax23 does mean a3x3. ■
(a) 1 - x223 = 3 1 - 12x243 = 1 - 1231x223 = - x6
exponent of 1
add exponents of a
(b) ax21ax23 = ax21a3x32 = a4x5 add exponents of x
Practice Exercises
Use Eqs. (1.3)–(1.6) to simplify the given expressions.
13 * 224 3424 3 * 24 (c) 13 * 523 = 3353 = 53
1. ax31 - ax22
12c25 2. 13cd22
1ry322 r2y6 r (d) r1y224 = ry8 = y2
■
1.4 Exponents and Unit Conversions
19
E X A M P L E 5 Exponents—beam deflection
In analyzing the amount a beam bends, the following simplification may be used. (P is the force applied to a beam of length L; E and I are constants related to the beam.)
1
a
PL
b
a
2
b
a
L
2
b
2 4EI 3 2
=
1 2
a
PL 4EI
b
a
2 3
b
a
L2 22
b
1 2PL
1L22
PL3
=
=
2132142142EI 48EI
1
In simplifying this expression, we combined exponents of L and divided out the 2 that
was a factor common to the numerator and the denominator.
■
ZERO AND NEGATIVE EXPONENTS If n = m in Eqs. (1.4), we have am>am = am - m = a0. Also, am>am = 1, since any nonzero quantity divided by itself equals 1. Therefore, for Eqs. (1.4) to hold for
m = n,
a0 = 1 1a ∙ 02
(1.7)
Equation (1.7) states that any nonzero expression raised to the zero power is 1. Zero exponents can be used with any of the operations for exponents.
E X A M P L E 6 Zero as an exponent
(a) 50 = 1
(b) 1 - 320 = 1
(c) - 1 - 320 = - 1
(d) 12x20 = 1
Practice Exercise 3. Evaluate: - 13x20
■ Although positive exponents are generally preferred in a final result, there are some cases in which zero or negative exponents are to be used. Also, negative exponents are very useful in some operations that we will use later.
(e) 1ax + b20 = 1
(f) 1a2b0c22 = a4c2
(g) 2t0 = 2112 = 2
b0 = 1
We note in illustration (g) that only t is raised to the zero power. If the quantity 2t were
raised to the zero power, it would be written as 12t20, as in part (d).
■
Applying both forms of Eq. (1.4) to the case where n 7 m leads to the definition of a negative exponent. For example, applying both forms to a2>a7, we have
a2 a7
=
a2 - 7
=
a-5
and
a2
1
1
a7 = a7 - 2 = a5
For these results to be equal, then a-5 = 1>a5. Thus, if we define
a-n
=
1 an
1a ∙ 02
(1.8)
then all the laws of exponents will hold for negative integers.
20
ChaPTER 1 Basic Algebraic Operations
■ Note carefully the difference in parts (d) and (e) of Example 7.
Practice Exercises
Simplify:
4.
- 70 c-3
13x2 -1 5. 2a-2
E X A M P L E 7 negative exponents
(a) 3-1 = 1 3
(b) 4-2
=
1 42
=
1 16
1 (c) a-3
=
a3
change signs of exponents
(d) 13x2-1 = 1 3x
(e)
3x -1
=
1 3a x b
=
3 x
(f)
a
a3 x
b
-2
=
1a32 -2 x-2
=
a-6 x-2
=
x2 a6
■
ORDER OF OPERATIONS
We have seen that the basic operations on numbers must be performed in a particular order. Since raising a number to a power is actually multiplication, it is performed before additions and subtractions, and in fact, before multiplications and divisions.
■ The use of exponents is taken up in more detail in Chapter 11.
order of operations 1. Operations within grouping symbols 2. Exponents 3. Multiplications and divisions (from left to right) 4. Additions and subtractions (from left to right)
Fig. 1.9
noTE →
E X A M P L E 8 using order of operations
(a) 8 - 1 - 122 - 21 - 322 = 8 - 1 - 2192 = 8 - 1 - 18 = -11
apply exponents first, then multiply, and then subtract
(b) 806 , 126.1 - 9.0922 = 806 , 117.0122 = 806 , 289.3401 = 2.79 subtract inside parentheses
first, then square the answer, and then divide
[In part (b), the significant digits retained from each intermediate step are underlined.] ■
noTE →
E X A M P L E 9 Even and odd powers
Using the meaning of a power of a number, we have
1 -222 = 1 -221 -22 = 4 1 -223 = 1 -221 -221 -22 = -8 1 - 224 = 16 1 - 225 = - 32 1 - 226 = 64 1 - 227 = - 128
[Note that a negative number raised to an even power gives a positive value, and a
negative number raised to an odd power gives a negative value.]
■
EVALUATING ALGEBRAIC EXPRESSIONS An algebraic expression is evaluated by substituting given values of the literal numbers in the expression and calculating the result. On a calculator, the x2 key is used to square numbers, and the ¿ or xy key is used for other powers.
To calculate the value of 20 * 6 + 200/5 - 34, we use the key sequence
20 * 6 + 200 , 5 - 3 ¿ 4
with the result of 79 shown in the display of Fig. 1.9. Note that calculators are programmed to follow the correct order of operations.
1.4 Exponents and Unit Conversions
21
Fig. 1.10
E X A M P L E 1 0 Evaluating an expression—free-fall distance
The distance (in ft) that an object falls in 4.2 s is found by substituting 4.2 for t in the expression 16.0t2 as shown below:
16.014.222 = 280 ft
The calculator result from the first line of Fig. 1.10 has been rounded off to two signifi-
cant digits, the accuracy of 4.2.
■
E X A M P L E 1 1 Evaluating an expression—length of a wire
A wire made of a special alloy has a length L (in m) given by L = a + 0.0115T3, where T (in °C) is the temperature (between -4°C and 4°C). To find the wire length for L for a = 8.380 m and T = -2.87°C, we substitute these values to get
L = 8.380 + 0.01151 - 2.8723 = 8.108 m
The calculator result from the second line of Fig. 1.10 has been rounded to the
nearest thousandth.
■
OPERATIONS wITH UNITS AND UNIT CONVERSIONS
Many problems in science and technology require us to perform operations on numbers with units. For multiplication, division, powers, or roots, whatever operation is performed on the numbers also is performed on the units. For addition and subtraction, only numbers with the same units can be combined, and the answer will have the same units as the numbers in the problem. Essentially, units are treated the same as any other algebraic symbol.
E X A M P L E 1 2 algebraic operations with units
(a) 12 ft214 lb2 = 8 ft # lb
the dot symbol represents multiplication
(b) 255 m + 121 m = 376 m
note that the units are not added
(c) 13.45 in.22 = 11.9 in.2
the unit is squared as well as the number
(d) a 65.0 mi b 13.52 h2 = 229 mi h
note that
mi h
*
h 1
=
mi 1
=
mi
(e)
8.48g 11.69m23 =
8.48g 11.6923m3
=
1.76 g/m3
the units are divided
8.75 mi 1 min 2 5280 ft
18.75 mi2 11 min22 15280 ft2
(f)
a 1.32 min2 b a 60 s b
a
1 mi
b
=
11.32 min22 13600 s22 11 mi2
the units min2 and mi both
cancel
= 9.72 ft/s2
■
Often, it is necessary to convert from one set of units to another. This can be accomplished by using conversion factors (for example, 1 in. = 2.54 cm). Several useful conversion factors are shown in Table 1.1.
Metric prefixes are sometimes attached to units to indicate they are multiplied by a given power of ten. Table 1.2 (on the next page) shows some commonly used prefixes.
Table 1.1 Conversion Factors
Length 1 in. = 2.54 cm (exact) 1 ft = 0.3048 m (exact) 1 km = 0.6214 mi 1 mi = 5280 ft (exact)
Volume/Capacity 1 ft3 = 28.32 L 1 L = 1.057 qt 1 gal = 3.785 L
Weight/Mass 1 lb = 453.6 g 1 kg = 2.205 lb 1 lb = 4.448 N
Energy/Power
1 Btu = 778.2 ft # lb 1 ft # lb = 1.356 J 1 hp = 550 ft # lb/s (exact)
1 hp = 746.0 W
22
ChaPTER 1 Basic Algebraic Operations
Table 1.2 Metric Prefixes
Prefix exa peta tera giga mega kilo hecto deca deci centi milli micro nano pico femto atto
Factor 1018 1015 1012 109 106 103 102 101 10-1 10-2 10-3 10-6 10-9 10-12 10-15 10-18
Symbol E P T G M k h da d c m m n p f a
■ See Appendix B for a description of the U.S. Customary and SI units, as well as a list of all the units used in this text and their symbols.
Practice Exercise 6. Convert 725 g/cm2 to lb/in.2
When a conversion factor is written in fractional form, the fraction has a value of 1
since the numerator and denominator represent the same quantity. For example,
1 in.
1 km
= 1 or
= 1. To convert units, we multiply the given number (including
2.54 cm
1000 m
its units) by one or more of these fractions placed in such a way that the units we wish
to eliminate will cancel and the units we wish to retain will remain in the answer. Since
we are multiplying the given number by fractions that have a value of 1, the original
quantity remains unchanged even though it will be expressed in different units.
E X A M P L E 1 3 Converting units
(a) The length of a certain smartphone is 13.8 cm. Convert this to inches.
13.8 cm
=
13.8 cm 1
*
1 in. 2.54 cm
=
113.82112 in. 11212.542
=
5.43 in.
original number
1
Notice that the unit cm appears in both the numerator and denominator and therefore cancels, leaving only the unit inches in the final answer.
(b) A car is traveling at 65.0 mi/h. Convert this speed to km/min (kilometers per minute). From Table 1.1, we note that 1 km = 0.6214 mi, and we know that 1 h = 60 min. Using these values, we have
mi 65.0
h
=
65.0 mi 1 h
*
1 km 0.6214 mi
*
1 h 60 min
=
165.02112112 km 11210.621421602 min
= 1.74 km/min
We note that the units mi and h appear in both numerator and denominator and therefore cancel out, leaving the units km and min. Also note that the 1’s and 60 are exact.
(c) The density of iron is 7.86 g/cm3 (grams per cubic centimeter). Express this density in kg/m3 (kilograms per cubic meter). From Table 1.2, 1 kg = 1000 g exactly, and 1 cm = 0.01 m exactly. Therefore,
g 7.86 cm3
=
7.86 g 1 cm3
*
1 kg 1000 g
*
1 cm 3
a
b
0.01 m
=
17.862 112 1132 kg 112 110002 10.0132 m3
=
17.862 kg 0.001 m3
=
7860 kg/m3
Here, the units g and cm3 are in both numerator and denominator and therefore cancel
out, leaving units of kg and m3. Also, all numbers are exact, except 7.86.
■
EXERCISES 1.4
In Exercises 1 and 2, make the given changes in the indicated examples of this section, and then simplify the resulting expression.
1. In Example 4(a), change 1 - x223 to 1 - x322. 2. In Example 6(d), change 12x20 to 2x0.
In Exercises 3–42, simplify the given expressions. Express results with positive exponents only.
3. x3x4
4. y2y7
5. 2b4b2
6. 3k5k
m5 7. m3 11. 1P224
23 15. a b
b 19. 18a20
23. 6-1
2x6 8. -x 12. 1x823
F 20 16. a b
t 20. - v0
24. - w-5
- n5 9. 7n9 13. 1aT2230
x2 4 17. a - 2 b 21. - 3x0
1 25. R-2
3s 10. s4 14. 13r223
33 18. a n3 b 22. - 1 - 220
1 26. - t-48
1.4 Exponents and Unit Conversions
23
27. 1 - t227 2v4
31. 12v24
28. 1 - y325 x2x3
32. 1x223
29.
-
L-3 L-5
1n224 33. 1n422
30. 2i40i-70 13t2 -1
34. - 3t-1
35. 1p0x2a-12-1
36. 13m-2n42-2
37. 1 - 8g-1s322
38. ax-21 - a2x23 15n2T 5
41. 3n-1T6
4x-1 -3 39. a a-1 b
1nRT -2232 42. R-2T32
2b2 -2 40. a - y5 b
In Exercises 43–50, evaluate the given expressions. In Exercises 45–50, all numbers are approximate.
43. 71 - 42 - 1 - 522
45. - 1 - 26.522 - 1 - 9.8523
3.071 - 0 -1.86 0 2
47. 1 - 1.8624 + 1.596
49.
2.381 - 10.722
-
254 1.173
44. 6 - 0 - 2 0 5 - 1 - 22182
46. - 0.7112 - 1 - 0 - 0.809 0 26
15.662 - 1 - 4.01724 48. 1.0441 - 3.682
50.
4.212.62
+
0.889 1.89 - 1.092
In Exercises 51–62, perform the indicated operations.
1 -1 51. Does a x-1 b represent the reciprocal of x?
52.
Does
a
0.2 - 5-1 10-2
b
0
equal
1?
Explain.
53. If a3 = 5, then what does a12 equal?
54. Is a-2 6 a-1 for any negative value of a? Explain.
# 55. If a is a positive integer, simplify 1xa x-a25. # 56. If a and b are positive integers, simplify 1 - ya - b ya + b22.
57. In developing the “big bang” theory of the origin of the universe, the expression 1kT> 1hc2 231GkThc22c arises. Simplify this expression.
58. In studying planetary motion, the expression 1GmM2 1mr2-11r-22 arises. Simplify this expression.
59. In designing a cam for a fire engine pump, the expression
p
a
r 2
3
ba
4 3pr2 b
is
used.
Simplify
this
expression.
60. For a certain integrated electric circuit, it is necessary to simplify
gM 12pfM2 -2
the expression
. Perform this simplification.
2pfC
61. If $2500 is invested at 4.2% interest, compounded quarterly, the amount in the account after 6 years is 250011 + 0.042>4224. Calculate this amount (the 1 is exact).
62. In designing a building, it was determined that the forces acting on an I beam would deflect the beam an amount (in cm), given by x11000 - 20x2 + x32 , where x is the distance (in m) from one 1850 end of the beam. Find the deflection for x = 6.85 m. (The 1000 and 20 are exact.)
63.
Calculate
the
value
of
a1
+
1n b
for
n
=
1, 10, 100, 1000
on
a
n
calculator. Round to four decimal places. (For even larger values of
n, the value will never exceed 2.7183. The limiting value is a num-
ber called e,which will be important in future chapters.)
64. For computer memory, the metric prefixes have an unusual meaning: 1 KB = 210 bytes, 1 MB = 210 KB, 1 GB = 210 MB, and 1 TB = 210 GB. How many bytes are there in 1 TB? (KB is kilo-
byte, MB is megabyte, GB is gigabyte, TB is terabyte)
In Exercises 65–68, perform the indicated operations and attach the correct units to your answers.
65. a28.2 ft b 19.81 s2 s
mi 66. a40.5 gal b 13.7 gal2
m
1 ft
60 s 2
67. a7.25 s2 b a 0.3048 m b a 1 min b
kg 1000 g 1 m 3
68.
a 238 m3 b a
1 kg
ba
b
100 cm
In Exercises 69–74, make the indicated conversions.
69. 15.7 qt to L
71. 245 cm2 to in.2
73.
65.2
m s
to
ft min
70. 7.50 W to hp
72. 85.7 mi2 to km2
74.
25.0
mi gal
to
km L
In Exercises 75–82, solve the given problems.
75. A laptop computer has a screen that measures 15.6 in. across its diagonal. Convert this to centimeters.
76. GPS satellites orbit the Earth at an altitude of about 12,500 mi. Convert this to kilometers.
77. A wastewater treatment plant processes 575,000 gal/day. Convert this to liters per hour.
78. Water flows from a fire hose at a rate of 85 gal/min. Convert this to liters per second.
79. The speed of sound is about 1130 ft/s. Change this to kilometers per hour.
80. A military jet flew at a rate of 7200 km/h. What is this speed in meters per second?
81. At sea level, atmospheric pressure is about 14.7 lb/in.2. Express this in pascals (Pa). Hint: A pascal is a N/m2 (see Appendix B).
82. The density of water is about 62.4 lb/ft3. Convert this to kilograms per cubic meter.
answers to Practice Exercises
1. a3x5
25c3 32c3 2. 32d2 = 9d2
3. - 1
4. - c3 5. a2 6x
6. 10.3 lb/in.2
24
ChaPTER 1 Basic Algebraic Operations
1.5 Scientific Notation
Meaning of Scientific Notation • Changing Numbers to and from Scientific Notation • Scientific Notation on a Calculator • Engineering notation
■ Television was invented in the 1920s and first used commercially in the 1940s. The use of fiber optics was developed in the 1950s. X-rays were discovered by Roentgen in 1895.
In technical and scientific work, we encounter numbers that are inconvenient to use in calculations. Examples are: radio and television signals travel at 30,000,000,000 cm/s; the mass of Earth is 6,600,000,000,000,000,000,000 tons; a fiber in a fiber-optic cable has a diameter of 0.000005 m; some X-rays have a wavelength of 0.000000095 cm. Although calculators and computers can handle such numbers, a convenient and useful notation, called scientific notation, is used to represent these or any other numbers.
A number in scientific notation is expressed as the product of a number greater than or equal to 1 and less than 10, and a power of 10, and is written as
P * 10k
where 1 … P 6 10 and k is an integer. (The symbol … means “is less than or equal to.”)
E X A M P L E 1 scientific notation (a) 34,000 = 3.4 * 10,000 = 3.4 * 104 (b) 6.82 = 6.82 * 1 = 6.82 * 100
(c)
0.00503
=
5.03 1000
=
5.03 103
=
5.03
*
10-3
between 1 and 10 ■
noTE →
From Example 1, we see how a number is changed from ordinary notation to scientific
notation. [The decimal point is moved so that only one nonzero digit is to its left. The
number of places moved is the power of 10 (k), which is positive if the decimal point is
moved to the left and negative if moved to the right.] To change a number from scientific
notation to ordinary notation, this procedure is reversed. The next two examples illustrate
these procedures.
E X A M P L E 2 Changing numbers to scientific notation (a) 34,000 = 3.4 * 104 (b) 6.82 = 6.82 * 100 (c) 0.00503 = 5.03 * 10-3
4 places to left
0 places
3 places to right
■
Practice Exercises 1. Change 2.35 * 10-3 to ordinary notation. 2. Change 235 to scientific notation.
E X A M P L E 3 Changing numbers to ordinary notation
(a) To change 5.83 * 106 to ordinary notation, we move the decimal point six places to the right, including additional zeros to properly locate the decimal point.
(b) To change 8.06 * 10-3 to ordinary notation, we must move the decimal point three places to the left, again including additional zeros to properly locate the decimal point.
5.83 * 106 = 5,830,000
8.06 * 10-3 = 0.00806
6 places to right
3 places to left ■
Scientific notation provides a practical way of handling very large or very small numbers. First, all numbers are expressed in scientific notation. Then the calculation can be done with numbers between 1 and 10, using the laws of exponents to find the power of ten of the result. Thus, scientific notation gives an important use of exponents.
1.5 Scientific Notation
25
■ See Exercise 43 of Exercises 1.1 for a brief note on computer data.
E X A M P L E 4 scientific notation in calculations—processing rate
The processing rate of a computer processing 803,000 bytes of data in 0.00000525 s is
5 - 1 -62 = 11
803,000 0.00000525
=
8.03 * 105 5.25 * 10-6
=
8.03 a 5.25 b
*
1011
=
1.53
*
1011 bytes/s
As shown, it is proper to leave the result (rounded off) in scientific notation. This method
is useful when using a calculator and then estimating the result. In this case, the estimate
is 18 * 1052 , 15 * 10-62 = 1.6 * 1011.
■
Another advantage of scientific notation is that it clearly shows the precise number of significant digits when the final significant digit is 0, making it unnecessary to use the “bar” notation introduced in Section 1.3.
E X A M P L E 5 scientific notation and significant digits—gravity
In determining the gravitational force between two stars 750,000,000,000 km apart, it is necessary to evaluate 750,000,000,0002. If 750,000,000,000 has three significant digits, we can show this by writing
750,000,000,0002 = 17.50 * 101122 = 7.502 * 102 * 11 = 56.3 * 1022
Since 56.3 is not between 1 and 10, we can write this result in scientific notation as
56.3 * 1022 = 15.63 * 102 110222 = 5.63 * 1023
■
We can enter numbers in scientific notation on a calculator, as well as have the calculator give results automatically in scientific notation. See the next example.
Fig. 1.11
E X A M P L E 6 scientific notation on a calculator—wavelength
The wavelength l (in m) of the light in a red laser beam can be found from the following calculation. Note the significant digits in the numerator.
l
=
3,000,000 4,740,000,000,000
=
3.00 * 106 4.74 * 1012
=
6.33
*
10-7 m
The key sequence is 3 EE 6 , 4.74 EE 12 ENTER . See Fig. 1.11. ■
Another commonly used notation, which is similar to scientific notation, is engineering notation. A number expressed in engineering notation is of the form
P * 10k
where 1"P * 1000 and k is an integral multiple of 3. Since the exponent k is a multiple of 3, the metric prefixes in Table 1.2 (Section 1.4) can be used to replace the power of ten. For example, an infrared wave that has a frequency of 850 * 109 Hz written in engineering notation can also be expressed as 850 GHz. The prefix giga (G) replaces the factor of 109.
Practice Exercise
3. Write 0.0000728 s in engineering notation and using the appropriate metric prefix.
E X A M P L E 7 Engineering notation and metric prefixes
Express each of the following quantities using engineering notation, and then replace the
power of ten with the appropriate metric prefix.
less than 1000
multiple of 3
(a) 48,000,000 Ω
= 48 * 106 Ω = 48 MΩ
(b) 0.00000036 m
= 360 * 10-9 m = 360 nm
(c) 1.3 * 10-4 A = 0.00013 A = 130 * 10-6 A = 130 mA
■
26
ChaPTER 1 Basic Algebraic Operations
EXERCISES 1.5
In Exercises 1 and 2, make the given changes in the indicated examples of this section, and then rewrite the number as directed.
1. In Example 3(b), change the exponent -3 to 3 and then write the number in ordinary notation.
2. In Example 5, change the exponent 2 to -1 and then write the result in scientific notation.
In Exercises 3–10, change the numbers from scientific notation to ordinary notation.
3. 4.5 * 104 4. 6.8 * 107 5. 2.01 * 10-3 6. 9.61 * 10-5
7. 3.23 * 100 8. 8 * 100
9. 1.86 * 10 10. 1 * 10-1
In Exercises 11–18, change the numbers from ordinary notation to scientific notation.
11. 4000
12. 56,000
15. 609,000,000 16. 10
13. 0.0087 17. 0.0528
14. 0.00074 18. 0.0000908
In Exercises 19–22, perform the indicated calculations using a calculator and by first expressing all numbers in scientific notation. Assume that all numbers are exact.
19. 28,000(2,000,000,000) 88,000
21. 0.0004
20. 50,000(0.006) 0.00003
22. 6,000,000
In Exercises 23–28, change the number from ordinary notation to engineering notation.
23. 35,600,000 26. 925,000,000,000
24. 0.0000056 27. 0.000000475
25. 0.0973 28. 370,000
In Exercises 29–32, perform the indicated calculations and then check the result using a calculator. Assume that all numbers are exact.
29. 2 * 10-35 + 3 * 10-34 31. 11.2 * 102923
30. 5.3 * 1012 - 3.7 * 1010 32. 12 * 10-162-5
In Exercises 33–40, perform the indicated calculations using a calculator. All numbers are approximate.
33. 1320(649,000)(85.3)
34. 0.0000569(3,190,000)
0.0732167102 35. 0.0013410.02312
0.00452 36. 2430197,1002
37.
13.642 * 10-82 12.736 * 1052
38.
17.309 * 10-122 5.984312.5036 * 10-202
13.69 * 10-72 14.61 * 10212
19.9 * 1072 11.08 * 101222
39. 0.0504
40. 13.603 * 10-52 120542
In Exercises 41–50, change numbers in ordinary notation to scientific notation or change numbers in scientific notation to ordinary notation. See Appendix B for an explanation of symbols used.
41. The average number of tweets per day on Twitter in 2015 was 500,000,000.
42. A certain laptop computer has 17,200,000,000 bytes of memory.
43. A fiber-optic system requires 0.000003 W of power.
44. A red blood cell measures 0.0075 mm across.
45. The frequency of a certain cell phone signal is 1,200,000,000 Hz.
46. The PlayStation 4 game console has a graphic processing unit that can perform 1.84 * 1012 floating point operations per second (1.84 teraflops). (Source: playstation.com.)
47. The Gulf of Mexico oil spill in 2010 covered more than 12,000,000,000 m2 of ocean surface.
48. A parsec, a unit used in astronomy, is about 3.086 * 1016 m. 49. The power of the signal of a laser beam probe is 1.6 * 10-12 W. 50. The electrical force between two electrons is about 2.4 * 10-43
times the gravitational force between them.
In Exercises 51–56, solve the given problems.
51. Write the following numbers in engineering notation and then replace the power of 10 with the appropriate metric prefix. (a) 2300 W (b) 0.23 W (c) 2,300,000 W (d) 0.00023 W
52. Write the following numbers in engineering notation and then replace the power of 10 with the appropriate metric prefix. (a) 8,090,000 Ω (b) 809,000 Ω (c) 0.0809 Ω
53. A googol is defined as 1 followed by 100 zeros. (a) Write this number in scientific notaion. (b) A googolplex is defined as 10 to the googol power. Write this number using powers of 10, and not the word googol. (Note the name of the Internet company.)
54. The number of electrons in the universe has been estimated at 1079. How many times greater is a googol than the estimated number of electrons in the universe? (See Exercise 53.)
55. The diameter of the sun, 1.4 * 109 m, is about 109 times the diameter of Earth. Express the diameter of Earth in scientific notation.
56. GB means gigabyte where giga means billion, or 109. Actually, 1 GB = 230 bytes. Use a calculator to show that the use of giga is a reasonable choice of terminology.
In Exercises 57–60, perform the indicated calculations.
57. A computer can do an addition in 7.5 * 10-15 s. How long does it take to perform 5.6 * 106 additions?
58. Uranium is used in nuclear reactors to generate electricity. About 0.000000039% of the uranium disintegrates each day. How much of 0.085 mg of uranium disintegrates in a day?
59. If it takes 0.078 s for a GPS signal traveling at 2.998 * 108 m/s to reach the receiver in a car, find the distance from the receiver to the satellite.
60. (a) Determine the number of seconds in a day in scientific notation. (b) Using the result of part (a), determine the number of seconds in a century (assume 365.24 days/year).
In Exercises 61–64, perform the indicated calculations by first expressing all numbers in scientific notation.
61. One atomic mass unit (amu) is 1.66 * 10-27 kg. If one oxygen atom has 16 amu (an exact number), what is the mass of 125,000,000 oxygen atoms?
1.6 Roots and Radicals
27
62. The rate of energy radiation (in W) from an object is found by evaluating the expression kT4, where T is the thermodynamic temperature. Find this value for the human body, for which k = 0.000000057 W/K4 and T = 303 K.
63. In a microwave receiver circuit, the resistance R of a wire 1 m long
# is given by R = k/d2, where d is the diameter of the wire. Find R
if k = 0.00000002196 Ω m2 and d = 0.00007998 m.
64. The average distance between the sun and Earth, 149,600,000 km, is called an astronomical unit (AU). If it takes light 499.0 s to travel 1 AU, what is the speed of light? Compare this with the speed of the GPS signal in Exercise 59.
answers to Practice Exercises 1. 0.00235 2. 2.35 * 102 3. 72.8 * 10-6 s, 72.8 ms
1.6 Roots and Radicals
Principal nth Root • Simplifying Radicals • Using a Calculator • Imaginary Numbers
■ Unless we state otherwise, when we refer to the root of a number, it is the principal root.
At times, we have to find the square root of a number, or maybe some other root of a
number, such as a cube root. This means we must find a number that when squared, or
cubed, and so on equals some given number. For example, to find the square root of 9, we must find a number that when squared equals 9. In this case, either 3 or -3 is an answer. Therefore, either 3 or -3 is a square root of 9 since 32 = 9 and 1 - 322 = 9.
To have a general notation for the square root and have it represent one number, we
define the principal square root of a to be positive if a is positive and represent it by 2a. This means 29 = 3 and not -3.
The general notation for the principal nth root of a is 2n a. (When n = 2, do not
write the 2 for n.) The 2 sign is called a radical sign.
E X A M P L E 1 Roots of numbers
(a) 22 (the square root of 2)
(b) 23 2 (the cube root of 2)
(c) 24 2 (the fourth root of 2)
(d) 27 6 (the seventh root of 6)
■
noTE →
[To have a single defined value for all roots (not just square roots) and to consider
only real-number roots, we define the principal nth root of a to be positive if a is posi-
tive and to be negative if a is negative and n is odd.] (If a is negative and n is even, the
roots are not real.)
E X A M P L E 2 Principal nth root (a) 2169 = 13 1 2169 ∙ -132
(b) - 264 = -8
(c) 23 27 = 3 since 33 = 27
(d) 20.04 = 0.2 since 0.22 = 0.04
odd
(e) - 24 256 = - 4 (f) 23 - 27 = - 3 (g) - 23 27 = - 1 + 32 = - 3
■
Fig. 1.12 Graphing calculator keystrokes: goo.gl/5HP7pF
■ Try this one on your calculator: 212345678987654321
The calculator evaluations of (b), (c), and (e) are shown in Fig. 1.12. The 1 key
is used for square roots and other roots are listed under the MATH key.
Another property of square roots is developed by noting illustrations such as 136 = 14 * 9 = 14 * 19 = 2 * 3 = 6. In general, this property states that the square root of a product of positive numbers is the product of their square roots.
2ab = 2a2b 1a and b positive real numbers2
(1.9)
This property is used in simplifying radicals. It is most useful if either a or b is a perfect square, which is the square of a rational number.
28
ChaPTER 1 Basic Algebraic Operations
E X A M P L E 3 Simplifying square roots (a) 28 = 2142122 = 2422 = 222
perfect squares simplest form
(b) 275 = 21252132 = 22523 = 523 (c) 24 * 102 = 24 2102 = 21102 = 20
(Note that the square root of the square of a positive number is that number.) ■
In order to represent the square root of a number exactly, use Eq. (1.9) to write it in simplest form. However, in many applied problems, a decimal approximation obtained from a calculator is acceptable.
E X A M P L E 4 Approximating a square root—rocket descent
After reaching its greatest height, the time (in s) for a rocket to fall h ft is found by evaluating 0.252h. Find the time for the rocket to fall 1260 ft.
Using a calculator, 0.2521260 = 8.9 s
The rocket takes 8.9 s to fall 1260 ft. The result from the calculator is rounded off to two
significant digits, the accuracy of 0.25 (an approximate number).
■
In simplifying a radical, all operations under a radical sign must be done before finding the root.
Practice Exercises Simplify:
1. 212 2. 236 + 144
E X A M P L E 5 More on simplifying square roots
(a) 216 + 9 = 225 first perform the addition 16 + 9
=5
noTE →
[However, 216 + 9 is not 216 + 29 = 4 + 3 = 7.]
(b) 222 + 62 = 24 + 36 = 240 = 24 210 = 2 210,
noTE →
[However, 222 + 62 is not 222 + 262 = 2 + 6 = 8.]
■
In defining the principal square root, we did not define the square root of a negative number. However, in Section 1.1, we defined the square root of a negative number to be an imaginary number. More generally, the even root of a negative number is an imaginary number, and the odd root of a negative number is a negative real number.
Practice Exercise
3. Is 23 - 8 real or imaginary? If it is real, evaluate it.
E X A M P L E 6 Imaginary roots and real roots
even
even
odd
2 - 64 is imaginary 24 - 243 is imaginary 23 - 64 = - 4 1a real number2 ■
A much more detailed coverage of roots, radicals, and imaginary numbers is taken up in Chapters 11 and 12.
EXERCISES 1.6
In Exercises 1–4, make the given changes in the indicated examples of this section and then solve the given problems.
1. In Example 2(b), change the square root to a cube root and then evaluate.
2. In Example 3(b), change 21252132 to 21152152 and explain whether or not this would be a better expression to use.
3. In Example 5(a), change the + to * and then evaluate.
1.7 Addition and Subtraction of Algebraic Expressions
29
4. In the first illustration of Example 6, place a - sign before the radical. Is there any other change in the statement?
In Exercises 5–38, simplify the given expressions. In each of 5–9 and 12–21, the result is an integer.
5. 249 9. - 264 13. 23 125 17. 1 2522
6. 2225 10. 20.25 14. 24 16 18. 1 23 3123
21. 1 - 24 5324 22. 275
7. - 2121 8. - 236
11. 20.09 12. - 2900
15. 24 81
16. - 25 - 32
19. 1 - 23 - 4723 20. 1 25 - 2325
23. 218
24. - 232
25. 21200 80
29. A 0 3 - 7 0
26. 250
27. 2284
30. 281 * 102 31. 23 - 82
2108 28.
2 32. 24 92
72 281 33.
1 - 322 249
25 25 - 243 34.
- 3 2144
35. 236 + 64
36. 225 + 144
37. 232 + 92
38. 282 - 42
In Exercises 39–46, find the value of each square root by use of a calculator. Each number is approximate.
39. 285.4 40. 23762
43. (a) 21296 + 2304 44. (a) 210.6276 + 2.1609 45. (a) 20.04292 - 0.01832 46. (a) 23.6252 + 0.6142
41. 20.8152 42. 20.0627
(b) 21296 + 22304 (b) 210.6276 + 22.1609 (b) 20.04292 - 20.01832 (b) 23.6252 + 20.6142
In Exercises 47–58, solve the given problems.
47. The speed (in mi/h) of a car that skids to a stop on dry pavement is often estimated by 224s, where s is the length (in ft) of the skid marks. Estimate the speed if s = 150 ft.
48. The resistance in an amplifier circuit is found by evaluating 2Z2 - X2. Find the resistance for Z = 5.362 Ω and X = 2.875 Ω.
49. The speed (in m/s) of sound in seawater is found by evaluating 2B>d for B = 2.18 * 109 Pa and d = 1.03 * 103 kg/m3. Find
this speed, which is important in locating underwater objects using sonar.
50. The terminal speed (in m/s) of a skydiver can be approximated by 240m, where m is the mass (in kg) of the skydiver. Calculate the terminal speed (after reaching this speed, the skydiver’s speed remains fairly constant before opening the parachute) of a 75-kg skydiver.
51. A TV screen is 52.3 in. wide and 29.3 in. high. The length of a
diagonal (the dimension used to describe it—from one corner to the opposite corner) is found by evaluating 2w2 + h2, where w is the width and h is the height. Find the diagonal.
52. A car costs $38,000 new and is worth $24,000 2 years later. The annual rate of depreciation is found by evaluating 10011 - 2V>C2, where C is the cost and V is the value after 2 years. At what rate did
the car depreciate? (100 and 1 are exact.)
53. A tsunami is a very high ocean tidal wave (or series of waves) often caused by an earthquake. An Alaskan tsunami in 1958 measured over 500 m high; an Asian tsunami in 2004 killed over 230,000 people; a tsunami in Japan in 2011 killed over 10,000 people. An equation that approximates the speed v (in m/s) of a tsunami is v = 2gd, where g = 9.8 m/s2 and d is the average depth (in m) of the ocean floor. Find v (in km/h) for d = 3500 m (valid for many parts of the Indian Ocean and Pacific Ocean).
54. The greatest distance (in km) a person can see from a height h (in m) above the ground is 21.27 * 104 h + h2. What is this distance for the pilot of a plane 9500 m above the ground?
55. Is it always true that 2a2 = a? Explain. 56. For what values of x is (a) x 7 1x, (b) x = 1x, and (c) x 6 1x?
57. A graphing calculator has a specific key sequence to find cube roots. Using a calculator, find 23 2140 and 23 - 0.214.
58. A graphing calculator has a specific key sequence to find nth roots. Using a calculator, find 27 0.382 and 27 - 382.
59. The resonance frequency f (in Hz) in an electronic circuit containing inductance L (in H) and capacitance C (in F) is given by
1
f=
. Find the resonance frequency if L = 0.250 H and
2p 2LC
C = 40.52 * 10-6 F.
60. In statistics, the standard deviation is the square root of the variance. Find the standard deviation if the variance is 80.5 kg2.
answers to Practice Exercises 1. 223 2. 625 3. real, -2
1.7 Addition and Subtraction of Algebraic Expressions
Algebraic Expressions • Terms • Factors • Polynomials • Similar Terms • Simplifying • symbols of grouping
Because we use letters to represent numbers, we can see that all operations that can be used on numbers can also be used on literal numbers. In this section, we discuss the methods for adding and subtracting literal numbers.
Addition, subtraction, multiplication, division, and taking powers or roots are known as algebraic operations. Any combination of numbers and literal symbols that results from algebraic operations is known as an algebraic expression.
30
ChaPTER 1 Basic Algebraic Operations
When an algebraic expression consists of several parts connected by plus signs and minus signs, each part (along with its sign) is known as a term of the expression. If a given expression is made up of the product of a number of quantities, each of these quantities, or any product of them, is called a factor of the expression.
CAUTION It is very important to distinguish clearly between terms and factors, because some operations that are valid for terms are not valid for factors, and conversely. Some of the common errors in handling algebraic expressions occur because these operations are not handled properly. ■
E X A M P L E 1 Terms and factors In the study of the motion of a rocket, the following algebraic expression may be used.
terms factors
gt2 - 2vt + 2s
This expression has three terms: gt2, -2vt, and 2s. The first term, gt2, has a factor of g
and two factors of t. Any product of these factors is also a factor of gt2. This means other
factors are gt, t2, and gt2 itself.
■
■ In Chapter 11, we will see that roots are equivalent to noninteger exponents.
E X A M P L E 2 Terms and factors
7a1x2 + 2y2 - 6x15 + x - 3y2 is an expression with terms 7a1x2 + 2y2 and
-6x15 + x - 3y2.
The term 7a1x2 + 2y2 has individual factors of 7, a, and 1x2 + 2y2, as well as prod-
ucts of these factors. The factor x2 + 2y has two terms, x2 and 2y.
The term -6x15 + x - 3y2 has factors 2, 3, x, and 15 + x - 3y2. The negative
sign in front can be treated as a factor of -1. The factor 5 + x - 3y has three terms, 5,
x, and -3y.
■
A polynomial is an algebraic expression with only nonnegative integer exponents on one or more variables, and has no variable in a denominator. The degree of a term is the sum of the exponents of the variables of that term, and the degree of the polynomial is the degree of the term of highest degree.
A multinomial is any algebraic expression of more than one term. Terms like 1>x and 1x can be included in a multinomial, but not in a polynomial. (Since 1>x = x-1, the exponent is negative.)
E X A M P L E 3 Polynomials
Some examples of polynomials are as follows:
(a) 4x2 - 5x + 3 (degree 2) (b) 2x6 - x (degree 6) (d) xy3 + 7x - 3 (degree 4) (add exponents of x and y) (e) - 6 (degree 0) 1 - 6 = - 6x02
(c) 3x
(degree 1)
From (c), note that a single term can be a polynomial, and from (e), note that a constant
can be a polynomial. The expressions in (a), (b), and (d) are also multinomials.
The expression x2 + 2y + 2 - 8 is a multinomial, but not a polynomial because
of the square root term.
■
A polynomial with one term is called a monomial. A polynomial with two terms is called a binomial, and one with three terms is called a trinomial. The numerical factor is called the numerical coefficient (or simply coefficient) of the term. All terms that differ at most in their numerical coefficients are known as similar or like terms. That is, similar terms have the same variables with the same exponents.
1.7 Addition and Subtraction of Algebraic Expressions
31
■ CAS (computer algebra system) calculators can display algebraic expressions and perform algebraic operations. The TI-89 graphing calculator is an example of such a calculator.
E X A M P L E 4 monomial, binomial, trinomial
(a) 7x4 is a monomial. The numerical coefficient is 7.
(b) 3ab - 6a is a binomial. The numerical coefficient of the first term is 3, and the numerical coefficient of the second term is -6. Note that the sign is attached to the
coefficient.
(c) 8cx3 - x + 2 is a trinomial. The coefficients of the first two terms are 8 and -1.
(d) x2y2 - 2x + 3y - 9 is a polynomial with four terms (no special name).
■
E X A M P L E 5 similar terms
(a) 8b - 6ab + 81b is a trinomial. The first and third terms are similar because they
differ only in their numerical coefficients. The middle term is not similar to the oth-
ers because it has a factor of a.
(b) 4x2 - 3x is a binomial. The terms are not similar since the first term has two factors
of x, and the second term has only one factor of x.
(c) 3x2y3 - 5y3x2 + x2 - 2y3 is a polynomial. The commutative law tells us that
x2y3 = y3x2, which means the first two terms are similar.
■
In adding and subtracting algebraic expressions, we combine similar terms into a single term. The simplified expression will contain only terms that are not similar.
E X A M P L E 6 simplifying expressions
(a) 3x + 2x - 5y = 5x - 5y add similar terms—result has unlike terms (b) 6a2 - 7a + 8ax cannot be simplified since there are no like terms.
(c) 6a + 5c + 2a - c = 6a + 2a + 5c - c commutative law
= 8a + 4c
add like terms
■
To group terms in an algebraic expression, we use symbols of grouping. In this text, we use parentheses, ( ); brackets, [ ]; and braces, 5 6. All operations that occur in the numerator or denominator of a fraction are implied to be inside grouping symbols, as well as all operations under a radical symbol.
CAUTION In simplifying an expression using the distributive law, to remove the symbols of grouping if a MINUS sign precedes the grouping, change the sign of EVERY term in the grouping, or if a plus sign precedes the grouping retain the sign of every term. ■
E X A M P L E 7 symbols of grouping
(a) 21a + 2x2 = 2a + 212x2 = 2a + 4x
use distributive law
(b) - 1 + a - 3c2 = 1 - 12 1 + a - 3c2 treat - sign as -1
= 1 -121 +a2 + 1 -121 -3c2
Practice Exercises Use the distributive law.
= - a + 3c note change of signs
1. 312a + y2 2. - 31 - 2r + s2
Normally, +a would be written simply as a.
■
32
ChaPTER 1 Basic Algebraic Operations
E X A M P L E 8 simplifying: signs before parentheses
+ sign before parentheses
(a) 3c + 12b - c2 = 3c + 2b - c = 2b + 2c
use distributive law
■ Note in each case that the parentheses are removed and the sign before the
parentheses is also removed.
2b = +2b
signs retained
- sign before parentheses
(b) 3c - 12b - c2 = 3c - 2b + c = -2b + 4c
2b = +2b
signs changed
(c) 3c - 1 -2b + c2 = 3c + 2b - c = 2b + 2c
use distributive law use distributive law
Practice Exercise 3. Simplify 2x - 314y - x2
signs changed
(d) y13 - y2 - 21y - 32 = 3y - y2 - 2y + 6
note the - 21 - 32 = + 6
= -y2 + y + 6
■
E X A M P L E 9 simplifying—machine part design
In designing a certain machine part, it is necessary to perform the following simplification.
1618 - x2 - 218x - x22 - 164 - 16x + x22 = 128 - 16x - 16x + 2x2 - 64 + 16x - x2
= 64 - 16x + x2
■
At times, we have expressions in which more than one symbol of grouping is to be
noTE → removed in the simplification. [Normally, when several symbols of grouping are to be removed, it is more convenient to remove the innermost symbols first.]
CAUTION One of the most common errors made is changing the sign of only the first term when removing symbols of grouping preceded by a minus sign. Remember, if the symbols are preceded by a minus sign, we must change the sign of every term. ■
TI-89 graphing calculator keystrokes for Example 11: goo.gl/sUCoav
E X A M P L E 1 0 several symbols of grouping
(a) 3ax - 3ax - 15s - 2ax24 = 3ax - 3ax - 5s + 2ax4
= 3ax - ax + 5s - 2ax
remove parentheses
= 5s
remove brackets
(b) 3a2b - 5 3a - 12a2b - a2 4 + 2b6 = 3a2b - 5 3a - 2a2b + a4 + 2b6
= 3a2b - 5a - 2a2b + a + 2b6
remove parentheses
= 3a2b - a + 2a2b - a - 2b
remove brackets
= 5a2b - 2a - 2b
remove braces
■
Calculators use only parentheses for grouping symbols, and we often need to use one set of parentheses within another set. These are called nested parentheses. In the next example, note that the innermost parentheses are removed first.
E X A M P L E 1 1 nested parentheses
2 - 13x - 215 - 17 - x222 = 2 - 13x - 215 - 7 + x22
= 2 - 13x - 10 + 14 - 2x2
= 2 - 3x + 10 - 14 + 2x = -x - 2
■
1.8 Multiplication of Algebraic Expressions
33
EXERCISES 1.7
In Exercises 1–4, make the given changes in the indicated examples of this section, and then solve the resulting problems.
1. In Example 6(a), change 2x to 2y. 2. In Example 8(a), change the sign before 12b - c2 from + to -. 3. In Example 10(a), change 3ax - 15s - 2ax24 to
31ax - 5s2 - 2ax4. 4. In Example 10(b), change 5 3a - 12a2b - a24 + 2b6 to
5a - 32a2b - 1a + 2b2 46.
In Exercises 5–51, simplify the given algebraic expressions.
5. 5x + 7x - 4x 7. 2y - y + 4x 9. 3t - 4s - 3t - s 11. 2F - 2T - 2 + 3F - T 13. a2b - a2b2 - 2a2b 15. 2p + 1p - 6 - 2p2
17. v - 17 - 9x + 2v2 19. 2 - 3 - 14 - 5a2
21. 1a - 32 + 15 - 6a2 23. - 1t - 2u2 + 13u - t2 25. 312r + s2 - 1 -5s - r2 27. - 716 - 3j2 - 21j + 42 29. - 314 - 6n2 - 1n - 324
6. 6t - 3t - 4t 8. - 4C + L - 6C 10. - 8a - b + 12a + b 12. x - 2y - 3x - y + z
14. - xy2 - 3x2y2 + 2xy2
16. 5 + 13 - 4n + p2
18.
- 2a
-
1 1b 2
-
a2
20. 2A + 1h - 22A2 - 32A
22. 14x - y2 - 1 -2x - 4y2
24. - 216x - 3y2 - 15y - 4x2
26. 31a - b2 - 21a - 2b2
28. - 15t + a22 - 213a2 - 2st2
30. - 31A - B2 - 1B - A24
31. 234 - 1t2 - 52 4
32.
-3c -3
-
2 3
1-a
-
42 d
33. - 23 - x - 2a - 1a - x24 34. - 23 -31x - 2y2 + 4y4
35. aZ - 33 - 1aZ + 424
36. 9v - 36 - 1 - v - 42 + 4v4
37. 5z - 58 - 34 - 12z + 1246 38. 7y - 5y - 32y - 1x - y246 39. 5p - 1q - 2p2 - 33q - 1p - q24 40. - 14 - 2LC2 - 3152LC - 72 - 162LC + 224 41. - 25 - 14 - x22 - 33 + 14 - x22 4 6 42. - 5 - 3 - 1x - 2a2 - b4 - 1a - x26 43. 5V2 - 16 - 12V2 + 322 44. - 2F + 2112F - 12 - 52 45. - 13t - 17 + 2t - 15t - 6222 46. a2 - 21x - 5 - 17 - 21a2 - 2x2 - 3x22 47. - 434R - 2.51Z - 2R2 - 1.512R - Z24
48. - 352.1e - 1.33 -f - 21e - 5f246
49. In determining the size of a V belt to be used with an engine, the expression 3D - 1D - d2 is used. Simplify this expression.
50. When finding the current in a transistor circuit, the expression i1 - 12 - 3i22 + i2 is used. Simplify this expression. (The numbers below the i’s are subscripts. Different subscripts denote dif-
ferent variables.)
51. Research on a plastic building material leads to
3 1B
+
4 3
a2
+
21B
-
2 3
a2
4
-
3 1B
+
4 3
a2
-
1B
-
2 3
a2
4.
Simplify this expression.
52. One car goes 30 km/h for t - 1 hours, and a second car goes 40 km/h for t + 2 hours. Find the expression for the sum of the
distances traveled by the two cars.
53. A shipment contains x hard drives with 4 terabytes of memory and x + 25 hard drives with 8 terabytes. Express the total number of terabytes of memory in the shipment as a variable expression and
simplify.
54. Each of two suppliers has 2n + 1 bundles of shingles costing $30 each and n - 2 bundles costing $20 each. How much more is the total value of the $30 bundles than the $20 bundles?
55. For the expressions 2x2 - y + 2a and 3y - x2 - b find (a) the sum, and (b) the difference if the second is subtracted from
the first.
56. For the following expressions, subtract the third from the sum of the first two: 3a2 + b - c3, 2c3 - 2b - a2, 4c3 - 4b + 3.
In Exercises 57–60, answer the given questions.
57. Is the following simplification correct? Explain.
2x - 3y + 5 - 14x - y + 32 = 2x - 3y + 5 - 4x - y + 3 = -2x - 4y + 8
58. Is the following simplification correct? Explain.
2a - 3b - 4c - 1 -5a + 3b - 2c2 = 2a - 3b - 4c + 5a - 3b - 2c = 7a - 6b - 6c
59. For any real numbers a and b, is it true that 0 a - b 0 = 0 b - a 0 ?
Explain. 60. Is subtraction associative? That is, in general, does 1a - b2 - c
equal a - 1b - c2? Explain.
answers to Practice Exercises 1. 6a + 3y 2. 6r - 3s 3. 5x - 12y
1.8 Multiplication of Algebraic Expressions
Multiplying Monomials • Products of Monomials and Polynomials • Powers of Polynomials
To find the product of two or more monomials, we multiply the numerical coefficients to find the numerical coefficient of the product, and multiply the literal numbers, remembering that the exponents may be combined only if the base is the same.
34
ChaPTER 1 Basic Algebraic Operations
E X A M P L E 1 multiplying monomials
(a) 3c51 - 4c22 = - 12c7
multiply numerical coefficients and add exponents of c
(b) 1 - 2b2y32 1 - 9aby52 = 18ab3y8 add exponents of same base
(c) 2xy1 - 6cx22 13xcy22 = - 36c2x4y3
■
noTE → [If a product contains a monomial that is raised to a power, we must first raise it to the indicated power before proceeding with the multiplication.]
E X A M P L E 2 Product containing power of a monomial
(a) - 3a12a2x23 = - 3a18a6x32 = - 24a7x3
(b) 2s31 - st42214s2t2 = 2s31s2t82 14s2t2 = 8s7t9
■
We find the product of a monomial and a polynomial by using the distributive law, which states that we multiply each term of the polynomial by the monomial. In doing so, we must be careful to give the correct sign to each term of the product.
E X A M P L E 3 Product of monomial and polynomial
Practice Exercises Perform the indicated multiplications.
1. 2a3b1 - 6ab22 2. - 5x2y312xy - y42
(a) 2ax13ax2 - 4yz2 = 2ax13ax22 + 12ax2 1 - 4yz2 = 6a2x3 - 8axyz
(b) 5cy21 - 7cx - ac2 = 15cy22 1 - 7cx2 + 15cy22 1 - ac2 = - 35c2xy2 - 5ac2y2 ■
It is generally not necessary to write out the middle step as it appears in the preceding example. We write the answer directly. For instance, Example 3(a) would appear as 2ax13ax2 - 4yz2 = 6a2x3 - 8axyz.
We find the product of two polynomials by using the distributive law. The result is that we multiply each term of one polynomial by each term of the other and add the results. Again we must be careful to give each term of the product its correct sign.
E X A M P L E 4 Product of polynomials
■ Note that, using the distributive law, 1x - 221x + 32 = 1x - 221x2 + 1x - 22132 leads to the same result.
1x - 221x + 32 = x1x2 + x132 + 1 -221x2 + 1 -22132
= x2 + 3x - 2x - 6 = x2 + x - 6
■
Finding the power of a polynomial is equivalent to using the polynomial as a factor the number of times indicated by the exponent. It is sometimes convenient to write the power of a polynomial in this form before multiplying.
E X A M P L E 5 Power of a polynomial
two factors
(a) 1x + 522 = 1x + 52 1x + 52 = x2 + 5x + 5x + 25 = x2 + 10x + 25
1.8 Multiplication of Algebraic Expressions
35
TI-89 graphing calculator keystrokes for Example 5: goo.gl/GKgiFl
Practice Exercises Perform the indicated multiplications. 3. 12s - 5t2 1s + 4t2 4. 13u + 2v22
(b) 12a - b23 = 12a - b2 12a - b2 12a - b2 the exponent 3 indicates three factors
= 12a - b2 14a2 - 2ab - 2ab + b22
= 12a - b2 14a2 - 4ab + b22
= 8a3 - 8a2b + 2ab2 - 4a2b + 4ab2 - b3
= 8a3 - 12a2b + 6ab2 - b3
■
CAUTION We should note that in Example 5(a) 1x + 522 is not equal to x2 + 25 because the term 10x is not included. We must follow the proper procedure and not
simply square each of the terms within the parentheses. ■
E X A M P L E 6 simplifying products—telescope lens
An expression used with a lens of a certain telescope is simplified as shown.
a1a + b22 + a3 - 1a + b2 12a2 - s22
= a1a + b2 1a + b2 + a3 - 12a3 - as2 + 2a2b - bs22
= a1a2 + ab + ab + b22 + a3 - 2a3 + as2 - 2a2b + bs2
= a3 + a2b + a2b + ab2 - a3 + as2 - 2a2b + bs2
= ab2 + as2 + bs2
■
EXERCISES 1.8
In Exercises 1–4, make the given changes in the indicated examples of this section, and then solve the resulting problems.
1. In Example 2(b), change the factor 1 - st422 to 1 - st423. 2. In Example 3(a), change the factor 2ax to -2ax. 3. In Example 4, change the factor 1x + 32 to 1x - 32. 4. In Example 5(b), change the exponent 3 to 2.
In Exercises 5–66, perform the indicated multiplications.
5. 1a22 1ax2
6. 12xy2 1x2y32
7. - a2c21a2cx32
8. - 2cs21 - 4cs22 11. i21Ri + 2i2
9. 12ax2221 - 2ax2 10. 6pq313pq222
12. 2x1 - p - q2
13. - 3s1s2 - 5t2
14. - 3b12b2 - b2 15. 5m1m2n + 3mn2 16. a2bc12ac - 3b2c2
17. 3M1 -M - N + 22
18. - 4c21 - 9gc - 2c + g22
19. xy1tx22 1x + y32
20. - 21 - 3st32 13s - 4t2
21. 1x - 321x + 52
22. 1a + 721a + 12
23. 1x + 5212x - 12
24. 14t1 + t22 12t1 - 3t22
25. 1y + 821y - 82
26. 1z - 421z + 42
27. 12a - b21 -2b + 3a2
28. 1 - 3 + 4w22 13w2 - 12
29. 12s + 7t213s - 5t2
30. 15p - 2q21p + 8q2
31. 1x2 - 12 12x + 52
32. 13y2 + 22 12y - 92
33. 1x - 2y - 421x - 2y + 42
34. 12a + 3b + 1212a + 3b - 12
35. 21a + 121a - 92
36. - 51y - 321y + 62
37. - 313 - 2T213T + 22
38. 2n1 - n + 5216n + 52
39. 2L1L + 1214 - L2
40. ax1x + 42 17 - x22
41. 13x - 722
42. 1x - 3y22
43. 1x1 + 3x222
44. 1 - 7m - 122 45. 1xyz - 222
46. 1 - 6x2 + b22
47. 21x + 822
48. 313R - 422
49. 12 + x2 13 - x2 1x - 12 50. 1 - c2 + 3x23
51. 3T1T + 2212T - 12
52. 3 1x - 2221x + 22 42
53. Let x = 3 and y = 4 to show that (a) 1x + y22 ∙ x2 + y2 and (b) 1x - y22 ∙ x2 - y2. ( ∙ means “does not equal”)
54. Explain how you would perform 1x + 325. Do not actually do the
operations.
55. By multiplication, show that 1x + y23 is not equal to x3 + y3.
56. By multiplication, show that 1x + y2 1x2 - xy + y22 = x3 + y3.
57. In finding the value of a certain savings account, the expression P11 + 0.01r22 is used. Multiply out this expression.
58. A savings account of $1000 that earns r% annual interest, compounded quarterly, has a value of 100011 + 0.0025r22 after 6 months. Perform the indicated multiplication.
59. A contractor is designing a rectangular room that will have a pool table. The length of the pool table is twice its width. The contractor wishes to have 5 ft of open space between each wall and the pool table. See Fig. 1.13. Express the area of the room in terms of the width w of the pool table. Then perform the indicated operations.
w 2w
Fig. 1.13
36
ChaPTER 1 Basic Algebraic Operations
60. The weekly revenue R (in dollars) of a flash drive manufacturer is given by R = xp, where x is the number of flash drives sold each week and p is the price (in dollars). If the price is given by the demand equation p = 30 - 0.01x, express the revenue in terms of x and simplify.
61. In using aircraft radar, the expression 12R - X22 - 1R2 + X22 arises. Simplify this expression.
62. In calculating the temperature variation of an industrial area, the expression 12T3 + 32 1T2 - T - 32 arises. Perform the indicated multiplication.
63. In a particular computer design containing n circuit elements, n2 switches are needed. Find the expression for the number of switches needed for n + 100 circuit elements.
64. Simplify the expression 1T2 - 10021T - 1021T + 102, which arises when analyzing the energy radiation from an object.
65. In finding the maximum power in part of a microwave transmiter circuit, the expression 1R1 + R222 - 2R21R1 + R22 is used. Multiply and simplify.
66. In determining the deflection of a certain steel beam, the expression 27x2 - 241x - 622 - 1x - 1223 is used. Multiply and simplify.
answers to Practice Exercises 1. - 12a4b3 2. - 10x3y4 + 5x2y7 3. 2s2 + 3st - 20t2 4. 9u2 + 12uv + 4v2
1.9 Division of Algebraic Expressions
Dividing Monomials • Dividing by a Monomial • Dividing One Polynomial by another
To find the quotient of one monomial divided by another, we use the laws of exponents and the laws for dividing signed numbers. Again, the exponents may be combined only if the base is the same.
E X A M P L E 1 Dividing monomials
(a)
3c7 c2
=
3c7 - 2
=
3c5
(b)
16x3y5 4xy2
=
16 1x3 - 12 1y5 - 22 4
=
4x2y3
(c)
- 6a2xy2 2axy4 =
-
a
6 2
b
a2 - 1x1 y4 - 2
1
=
- 3a y2
divide coefficients
subtract exponents
As shown in illustration (c), we use only positive exponents in the final result unless there
are specific instructions otherwise.
■
■ This is an identity and is valid for all values of a and b, and all values of c except
zero (which would make it undefined).
From arithmetic, we may show how a multinomial is to be divided by a monomial.
When adding fractions
1say
2 7
and
732,
we
have
2 7
+
3 7
=
2
+ 7
3 .
Looking at this from right to left, we see that the quotient of a multinomial divided by a monomial is found by dividing each term of the multinomial by the monomial and adding the results. This can be shown as
a
+ c
b
=
a c
+
b c
CAUTION
c a
+
c . ■
b
Be careful: Although
a
+ c
b
=
a c
+
b c,
we must
note that
a
c +
b
is
not
E X A M P L E 2 Dividing by a monomial
(a)
4a2 + 2a
8a
=
4a2 2a
+
8a 2a
=
2a
+
4
(b)
4x3y
-
8x3y2 2x2y
+
2x2y
=
4x3y 2x2y
-
8x3y2 2x2y
+
2x2y 2x2y
each term of numerator divided by denominator
= 2x - 4xy + 1
1.9 Division of Algebraic Expressions
37
■ Until you are familiar with the method, it is recommended that you do write out the
middle steps.
Practice Exercise 4ax2 - 6a2 x
1. Divide: 2ax
noTE →
We usually do not write out the middle step as shown in these illustrations. The divisions
of the terms of the numerator by the denominator are usually done by inspection (men-
tally), and the result is shown as it appears in the next example.
[Note carefully the last term 1 of the result. When all factors of the numerator are the
same as those in the denominator, we are dividing a number by itself, which gives a
result of 1.]
■
E X A M P L E 3 Dividing by a monomial—irrigation pump
2p + v2d + 2ydg
The expression
is used when analyzing the operation of an irrigation
2dg
pump. Performing the indicated division, we have
2p
+
v2d + 2dg
2ydg
=
p dg
+
v2 2g
+
y
■
■ This is similar to long division of numbers.
DIVISION OF ONE POLyNOMIAL By ANOTHER To divide one polynomial by another, use the following steps.
1. Arrange the dividend (the polynomial to be divided) and the divisor in descending powers of the variable.
2. Divide the first term of the dividend by the first term of the divisor. The result is the first term of the quotient.
3. Multiply the entire divisor by the first term of the quotient and subtract the product from the dividend.
4. Divide the first term of this difference by the first term of the divisor. This gives the second term of the quotient.
5. Multiply this term by the entire divisor and subtract the product from the first difference.
6. Repeat this process until the remainder is zero or of lower degree than the divisor.
7.
Express the answer in the form quotient
+
remainder .
divisor
E X A M P L E 4 Dividing one polynomial by another
Perform the division 16x2 + x - 22 , 12x - 12.
a This division can also be indicated in the fractional form
6x2 + x 2x -
1
2 .
b
We set up the division as we would for long division in arithmetic. Then, following the procedure outlined above, we have the following:
■ The answer to Example 4 can be checked by showing 12x - 12 13x + 22 = 6x2 + x - 2.
Practice Exercise 2. Divide: 16x2 + 7x - 32 , 13x - 12
3x + 2
2x - 1 ∙ 6x2 + x - 2
6x2 2x
subtract
6x2 - 6x2 = 0 x - 1 -3x2 = 4x
6x2 - 3x 4x - 2 4x - 2
divide first term of dividend by first term of divisor
3x12x - 12
0
The remainder is zero and the quotient is 3x + 2. Therefore, the answer is
3x
+
2
+
0 2x -
, or simply 3x 1
+
2
■
38
ChaPTER 1 Basic Algebraic Operations
TI-89 graphing calculator keystrokes for Example 5: goo.gl/WtS19R
■ The answer to Example 5 can be checked by showing
14x2 - 12 12x + 12 + 12x + 42 = 8x3 + 4x2 + 3
E X A M P L E 5 quotient with a remainder
Perform
the
division
8x3 + 4x2 + 4x2 - 1.
3
Because
there
is
no
x-term
in
the
dividend,
we
should leave space for any x-terms that might arise (which we will show as 0x).
2x + 1
divisor
4x2 - 1 ∙ 8x3 + 4x2 + 0x + 3
8x3
- 2x
0x = 1 -2x2 = 2x
4x2 + 2x + 3
4x2
-1
2x + 4
dividend subtract
8x3 4x2 = 2x
subtract
4x2 4x2 = 1
remainder
Because the degree of the remainder 2x + 4 is less than that of the divisor, the long-
division process is complete and the answer is 2x
+
1
+
2x 4x 2
+ -
41.
■
EXERCISES 1.9
In Exercises 1–4, make the given changes in the indicated examples of this section and then perform the indicated divisions.
1. In Example 1(c), change the denominator to - 2a2xy5. 2. In Example 2(b), change the denominator to 2xy2. 3. In Example 4, change the dividend to 6x2 - 7x + 2. 4. In Example 5, change the sign of the middle term of the numerator
from + to - .
In Exercises 5–24, perform the indicated divisions.
8x3y2 5. - 2xy
- 18b7c3 6. bc2
- 16r3t5 7. - 4r5t
51mn5 8. 17m2n2
115x2y2 12xz2 9.
10xy
15sT2 18s2T32
10.
10s3T 2
14a32 12x22 11. 14ax22
12a2b 12. 13ab222
3a2x + 6xy 13.
3x
2m2n - 6mn
14.
- 2m
3rst - 6r2st2 15.
3rs
- 5a2n - 10an2 16.
5an
4pq3 + 8p2q2 - 16pq5
17.
4pq2
18. a2x1x22 + ax13 - ax1 ax1
2pfL - pfR2 19.
pfR
91aB24 - 6aB4
20.
- 3aB3
- 7a2b + 14ab2 - 21a3
21.
14a2b2
2xn + 2 + 4axn
22.
2xn
6y2n - 4ayn + 1
23.
2yn
3a1F + T2b2 - 1F + T2
24.
a1F + T2
In Exercises 25–44, perform the indicated divisions. Express the answer as shown in Example 5 when applicable.
25. 1x2 + 9x + 202 , 1x + 42 26. 1x2 + 7x - 182 , 1x - 22 27. 12x2 + 7x + 32 , 1x + 32 28. 13t2 - 7t + 42 , 1t - 12
29. 1x2 - 3x + 22 , 1x - 22 30. 12x2 - 5x - 72 , 1x + 12 31. 1x - 14x2 + 8x32 , 12x - 32 32. 16 + 7y + 6y22 , 12y + 12 33. 14Z2 - 5Z - 72 , 14Z + 32 34. 16x2 - 5x - 92 , 1 - 4 + 3x2
x3 + 3x2 - 4x - 12
35.
x+2
2a4 + 4a2 - 16
37.
a2 - 2
y3 + 27 39. y + 3
x2 - 2xy + y2
41.
x-y
t3 - 8 43. t2 + 2t + 4
3x3 + 19x2 + 13x - 20
36.
3x - 2
6T3 + T2 + 2 38. 3T2 - T + 2
D3 - 1 40. D - 1
3r2 - 5rR + 2R2
42.
r - 3R
a4 + b4 44. a2 - 2ab + 2b2
In Exercises 45–56, solve the given problems.
45. When 2x2 - 9x - 5 is divided by x + c, the quotient is 2x + 1.
Find c.
46. When 6x2 - x + k is divided by 3x + 4, the remainder is zero.
47.
Find k. By division show that
x4 x
+ +
1 1
is
not
equal
to
x3.
48.
By division show that
x3 x
+ +
y3 is not equal to x2 y
+
y2.
49. If a gas under constant pressure has volume V1 at temperature T1
(in kelvin), then the new volume V2 when the temperature changes
from
T1
to
T2
is
given
by
V2
=
V1 a 1
+
T2
T1
T1 b .
Simplify
the
right-hand side of this equation.
1.10 Solving Equations
39
50. The area of a certain rectangle can be represented by 6x2 + 19x + 10. If the length is 2x + 5, what is the width? (Divide the area by the length.)
51. In the optical theory dealing with lasers, the following expression
8A5 + 4A3m2E2 - Am4E4
arises:
8A4
. (m is the Greek letter mu.)
Simplify this expression.
6 Æ
52. In finding the total resistance of
the resistors shown in Fig. 1.14,
the following expression is used.
R2
6R1 + 6R2 + R1R2
R1
6R1R2
Simplify this expression.
Fig. 1.14
53. When analyzing the potential energy associated with gravitational
GMm31R + r2 - 1R - r24
forces, the expression
arises.
2rR
Simplify this expression.
54. A computer model shows that the temperature change T in a certain
freezing
unit
is
found
by
using
the
expression
3T 3
T
8T 2 -2
+
8 .
Perform the indicated division.
55. In analyzing the displacement of a certain valve, the expression
s2
s4
2s +
4
2
is
used.
Find
the
reciprocal
of
this
expression
and
then perform the indicated division.
56. In analyzing a rectangular computer image, the area and width of
the image vary with time such that the length is given by the
expression
2t3
+
94t2 2t
+
290t 100
+
500 .
By
performing
the
indi-
cated division, find the expression for the length.
answers to Practice Exercises 1. 2x - 3a 2. 2x + 3
1.10 Solving Equations
Types of Equations • Solving Basic Types of Equations • Checking the Solution • First Steps • Ratio and Proportion
In this section, we show how algebraic operations are used in solving equations. In the following sections, we show some of the important applications of equations.
An equation is an algebraic statement that two algebraic expressions are equal. Any value of the unknown that produces equality when substituted in the equation is said to satisfy the equation and is called a solution of the equation.
E X A M P L E 1 Valid values for equations
The equation 3x - 5 = x + 1 is true only if x = 3. Substituting 3 for x in the equation, we have 3132 - 5 = 3 + 1, or 4 = 4; substituting x = 2, we have 1 = 3, which is
not correct.
This equation is valid for only one value of the unknown. An equation valid only for
certain values of the unknown is a conditional equation. In this section, nearly all equa-
tions we solve will be conditional equations that are satisfied by only one value of the
unknown.
■
E X A M P L E 2 Identity and contradiction
(a) The equation x2 - 4 = 1x - 22 1x + 22 is true for all values of x. For example, substituting x = 3 in the equation, we have 32 - 4 = 13 - 22 13 + 22, or 5 = 5. Substituting x = -1, we have 1 -122 - 4 = 1 -1 - 22 1 -1 + 22, or -3 = -3.
An equation valid for all values of the unknown is an identity.
(b) The equation x + 5 = x + 1 is not true for any value of x. For any value of x we
try, we find that the left side is 4 greater than the right side. Such an equation is
called a contradiction.
■
■ Equations can be solved on most graphing calculators. An estimate (or guess) of the answer may be required to find the solution. See Exercises 47 and 48.
To solve an equation, we find the values of the unknown that satisfy it. There is one basic rule to follow when solving an equation:
Perform the same operation on both sides of the equation.
We do this to isolate the unknown and thus to find its value.
40
ChaPTER 1 Basic Algebraic Operations
By performing the same operation on both sides of an equation, the two sides remain equal. Thus,
we may add the same number to both sides, subtract the same number from both sides, multiply both sides by the same number (not zero), or divide both sides by the same number (not zero).
■ The word algebra comes from Arabic and means “a restoration.” It refers to the fact that when a number has been added to one side of an equation, the same number must be added to the other side to maintain equality.
E X A M P L E 3 Basic operations used in solving
In each of the following equations, we may isolate x, and thereby solve the equation, by performing the indicated operation.
x - 3 = 12
add 3 to both sides
x - 3 + 3 = 12 + 3
x + 3 = 12
subtract 3 from both sides
x + 3 - 3 = 12 - 3
x = 12
3
multiply both sides by 3
3a x b = 31122 3
3x = 12
divide both sides by 3
3x 12 =
33
x = 15
x=9
x = 36
x=4
noTE → [Each solution should be checked by substitution in the original equation.]
■
■ Note that the solution generally requires a combination of basic operations.
E X A M P L E 4 operations used for solution; checking
Solve the equation 2t - 7 = 9. We are to perform basic operations to both sides of the equation to finally isolate t on
one side. The steps to be followed are suggested by the form of the equation.
2t - 7 = 9 2t - 7 + 7 = 9 + 7
original equation add 7 to both sides
2t = 16
combine like terms
2t 16 =
22
t=8
divide both sides by 2 simplify
Therefore, we conclude that t = 8. Checking in the original equation, we have 2182 - 7 ≟ 9, 16 - 7 ≟ 9, 9 = 9
The solution checks.
■
■ With simpler numbers, many basic steps are done by inspection and not actually written down.
Practice Exercises Solve for x. 1. 3x + 4 = x - 6 2. 215 - x2 = x - 8
E X A M P L E 5 First remove parentheses
Solve the equation x - 7 = 3x - 16x - 82.
x - 7 = 3x - 6x + 8 x - 7 = -3x + 8 4x - 7 = 8
parentheses removed x-terms combined on right 3x added to both sides
4x = 15
x
=
15 4
7 added to both sides both sides divided by 4
Checking
in
the
original
equation,
we
obtain
(after
simplifying)
-
13 4
=
- 143.
■
CAUTION Note that we always check in the original equation. This is done since errors may have been made in finding the later equations. ■
1.10 Solving Equations
41
■ Many other types of equations require more advanced methods for solving. These
are considered in later chapters.
From these examples, we see that the following steps are used in solving the basic equations of this section.
Procedure for Solving Equations 1. Remove grouping symbols (distributive law). 2. Combine any like terms on each side (also after step 3). 3. Perform the same operations on both sides until x = solution is obtained. 4. Check the solution in the original equation.
noTE → [If an equation contains numbers not easily combined by inspection, the best procedure is to first solve for the unknown and then perform the calculation.]
Fig. 1.15
Graphing calculator keystrokes: goo.gl/5wWTnp
E X A M P L E 6 First solve for unknown—circuit current
When finding the current i (in A) in a certain radio circuit, the following equation and solution are used.
0.0595 - 0.525i - 8.851i + 0.003162 = 0
0.0595 - 0.525i - 8.85i - 8.8510.003162 = 0
1 -0.525 - 8.852i = 8.8510.003162 - 0.0595
8.8510.003162 - 0.0595
i=
-0.525 - 8.85
= 0.00336 A
note how the above procedure is followed
evaluate
The calculator solution of this equation, using the Solver feature, is shown in Fig. 1.15.
When doing the calculation indicated above in the solution for i, be careful to group the
numbers in the denominator for the division. Also, be sure to round off the result as
shown above, but do not round off values before the final calculation.
■
■ In general, the proportion a = c is bd
equivalent to the equation ad = bc.
Therefore, x
=
3 can be rewritten as
84
4x = 24 in order to remove the fractions.
Practice Exercise
3. If the ratio of 2 to 5 equals the ratio of x to 30, find x.
RATIO AND PROPORTION
The quotient a>b is also called the ratio of a to b. An equation stating that two ratios are equal is called a proportion. Because a proportion is an equation, if one of the numbers is unknown, we can solve for its value as with any equation. Usually, this is done by noting the denominators and multiplying each side by a number that will clear the fractions.
E X A M P L E 7 Ratio
If the ratio of x to 8 equals the ratio of 3 to 4, we have the proportion
x3 =
84
We can solve this equation by multiplying both sides by 8. This gives
8a x b = 8a 3 b , or x = 6
8
4
Substituting x
=
6
into
the
original
proportion
gives
the
proportion
6 8
=
34. Because these
ratios are equal, the solution checks.
■
42
ChaPTER 1 Basic Algebraic Operations
■ Generally, units of measurement will not be shown in intermediate steps. The proper units will be shown with the data and final result.
■ If the result is required to be in feet, we have the following change of units (see Section 1.4):
6.08 ma 1 ft b = 19.9 ft 0.3048 m
E X A M P L E 8 Proportion—roof truss
The supports for a roof are triangular trusses for which the longest side is 8>5 as long as the shortest side for all of the trusses. If the shortest side of one of the trusses is 3.80 m, what is the length of the longest side of that truss?
If we label the longest side L, since the ratio of sides is 8>5, we have
L8 =
3.80 5
3.80
a
L 3.80
b
=
3.80
a
8 5
b
L = 6.08 m
Checking, we note that 6.08>3.80 = 1.6 and 8>5 = 1.6. The solution checks.
■
The meanings of ratio and proportion (particularly ratio) will be of importance when studying trigonometry in Chapter 4. A detailed discussion of ratio and proportion is found in Chapter 18. A general method of solving equations involving fractions, such as we found in Examples 7 and 8, is given in Chapter 6.
EXERCISES 1.10
In Exercises 1–4, make the given changes in the indicated examples of this section and then solve the resulting problems.
1. In Example 3, change 12 to - 12 in each of the four illustrations and then solve.
2. In Example 4, change 2t - 7 to 7 - 2t and then solve. 3. In Example 5, change 16x - 82 to 18 - 6x2 and then solve. 4. In Example 8, change 8>5 to 7>4 and then solve.
In Exercises 5–44, solve the given equations.
5. x - 2 = 7
8. s + 6 = -3
y-8
11.
=4
3
14. 2x = 12
6. x - 4 = - 1
9.
-
t 2
=
5
12. 7 - r = 3 6
15. 5t + 9 = - 1
7. x + 5 = 4 x
10. - 4 = 2
13. 4E = -20 16. 5D - 2 = 13
17. 5 - 2y = -3 18. - 5t + 8 = 18 19. 3x + 7 = x
20. 6 + 4L = 5 - 3L
21. 213q + 42 = 5q
22. 314 - n2 = -n
23. - 1r - 42 = 6 + 2r
24. - 1x + 22 + 5 = 5x
25. 81y - 52 = - 2y
26. 417 - F2 = - 7 28. 1.5x - 0.31x - 42 = 6
27. 0.1x - 0.51x - 22 = 2 29. - 4 - 311 - 2p2 = -7 + 2p
30. 3 - 612 - 3t2 = t - 5
4x - 21x - 42
31.
=8
3
- 517 - 3x2 + 2
32. 2x =
4
33. 0 x 0 - 9 = 2
34. 2 - 0 x 0 = 4 35. 0 2x - 3 0 = 5 36. 0 7 - x 0 = 1
In Exercises 37–44, all numbers are approximate. 37. 5.8 - 0.31x - 6.02 = 0.5x 38. 1.9t = 0.514.0 - t2 - 0.8
39. - 0.241C - 0.502 = 0.63 x 17
41. = 2.0 6.0 165 13V
43. = 223 15
40. 27.515.17 - 1.44x2 = 73.4
3.0 R 42. =
7.0 42
276x 1360
44.
=
17.0 46.4
In Exercises 45–56, solve the given problems.
45. Identify each of the following equations as a conditional equation,
an identity, or a contradiction.
(a) 2x + 3 = 3 + 2x
(b) 2x - 3 = 3 - 2x
46. Are there any values of a for which the equation 2x + a = 2x
results in a conditional equation? Explain why or why not.
47. Solve the equation of Example 5 by using the Equation Solver of a graphing calculator.
48. Solve the equation of Example 6 by using the Equation Solver of a graphing calculator.
49. To find the amount of a certain investment of x dollars, it is necessary to solve the equation 0.03x + 0.0612000 - x2 = 96. Solve for x.
50. In finding the rate v (in km/h) at which a polluted stream is flowing, the equation 1515.5 + v2 = 2415.5 - v2 is used. Find v.
51. In finding the maximum operating temperature T (in °C) for a computer integrated circuit, the equation 1.1 = 1T - 762 >40 is used. Find the temperature.
52. To find the voltage V in a circuit in a TV remote-control unit, the equation 1.12V - 0.67110.5 - V2 = 0 is used. Find V.
53. In blending two gasolines of different octanes, in order to find the
number n of gallons of one octane needed, the equation 0.14n + 0.0612000 - n2 = 0.09120002 is used. Find n, given that 0.06 and 0.09 are exact and the first zero of 2000 is
significant.
1.11 Formulas and Literal Equations
43
54. In order to find the distance x such that the weights are bal-
anced on the lever shown in Fig. 1.16, the equation 21013x2 = 55.3x + 38.518.25 - 3x2 must be solved. Find x. (3 is exact.)
210 N 55.3 N
38.5 N
3x x 8.25 m
Fig. 1.16
55.
Tchhaerg2e0d1630N@kisWsan# hLbeaatfteerlye.ctHriocwcamr uccahn
travel 107 electricity
mi (in
oknWa#
fully h) is
required to drive this car on a 350-mi trip? Assume all numbers are
exact and round your answer to a whole number. (Source: www
.nissanusa.com.)
56. An athlete who was jogging and wearing a Fitbit found that she burned 250 calories in 20 minutes. At that rate, how long will it take her to burn 400 calories? Assume all numbers are exact.
answers to Practice Exercises 1. -5 2. 6 3. 12
1.11 Formulas and Literal Equations
Formulas • Literal Equations • Subscripts • solve for symbol before substituting numerical values
■ Einstein published his first paper on relativity in 1905.
noTE →
An important application of equations is in the use of formulas that are found in geometry
and nearly all fields of science and technology. A formula (or literal equation) is an
equation that expresses the relationship between two or more related quantities. For example, Einstein’s famous formula E = mc2 shows the equivalence of energy E to the
mass m of an object and the speed of light c.
We can solve a formula for a particular symbol just as we solve any equation. [That is, we isolate the required symbol by using algebraic operations on literal numbers.]
E X A M P L E 1 solving for symbol in formula—Einstein
In Einstein’s formula E = mc2, solve for m.
E c2 = m
E m = c2
divide both sides by c2 switch sides to place m at left
The required symbol is usually placed on the left, as shown.
■
■ The subscript0 makes v0 a different literal symbol from v. (We have used subscripts in a few of the earlier exercises.)
E X A M P L E 2 symbol with subscript in formula—velocity
A formula relating acceleration a, velocity v, initial velocity v0, and time is v = v0 + at. Solve for t.
v - v0 = at
v0 subtracted from both sides
t
=
v
a
v0
both sides divided by a and then sides switched
■
TI-89 graphing calculator keystrokes for Example 3: goo.gl/1fYsOi
CAUTION Be careful. Just as subscripts can denote different literal numbers, a capital letter and the same letter in lowercase are different literal numbers. In this example, W and w are different literal numbers. This is shown in several of the exercises in this section. ■
E X A M P L E 3 symbol in capital and in lowercase—forces on a beam
L1wL + 2P2
In the study of the forces on a certain beam, the equation W =
8
is used.
Solve for P.
8L1wL + 2P2
8W =
8
8W = L1wL + 2P2
multiply both sides by 8 simplify right side
8W = wL2 + 2LP
remove parentheses
8W - wL2 = 2LP
subtract wL2 from both sides
8W - wL2
P = 2L
divide both sides by 2L and switch sides
■
44
ChaPTER 1 Basic Algebraic Operations
E X A M P L E 4 Formula with groupings—temperature and volume
The effect of temperature on measurements is important when measurements must be made with great accuracy. The volume V of a special precision container at temperature T in terms of the volume V0 at temperature T0 is given by V = V031 + b1T - T02 4, where b depends on the material of which the container is made. Solve for T.
Because we are to solve for T, we must isolate the term containing T. This can be done by first removing the grouping symbols.
Practice Exercises Solve for the indicated letter. Each comes from the indicated area of study. 1. u = kA + l, for l (robotics) 2. P = n1p - c2, for p (economics)
noTE →
V = V031 + b1T - T02 4 original equation
V = V031 + bT - bT04
remove parentheses
V = V0 + bTV0 - bT0V0 remove brackets
V - V0 + bT0V0 = bTV0 T = V - V0 + bT0V0 bV0
subtract V0 and add bT0V0 to both sides divide both sides by bV0 and switch sides ■
[To determine the values of any literal number in an expression for which we know
values of the other literal numbers, we should first solve for the required symbol and then
evaluate.]
■ Note that copper melts at about 1100°C.
E X A M P L E 5 solve for symbol before substituting—volume of sphere
The volume V (in mm3) of a copper sphere changes with the temperature T (in °C)
according to V = V0 + V0bT, V0 = 6715 mm3 and b = 5.10
w* h1e0re-5V>0°Cis.
the volume at 0°C. For Evaluate T for V = 6908
a given mm3.
sphere,
We first solve for T and then substitute the given values.
V = V0 + V0bT V - V0 = V0bT
T = V - V0 V0b
Now substituting, we have 6908 - 6715
T = 167152 15.10 * 10-52
= 564°C
■
EXERCISES 1.11
In Exercises 1–4, solve for the given letter from the indicated example of this section.
1. For the formula in Example 2, solve for a. 2. For the formula in Example 3, solve for w. 3. For the formula in Example 4, solve for T0. 4. For the formula in Example 5, solve for b. (Do not evaluate.)
In Exercises 5–42, each of the given formulas arises in the technical or scientific area of study shown. Solve for the indicated letter.
5. E = IR, for R (electricity) 6. PV = nRT, for T (chemistry) 7. rL = g2 - g1, for g1 (surveying) 8. W = SdT - Q, for Q (air conditioning)
nTWL
9. B =
, for n (construction management)
12
10. P = 2pTf, for T (mechanics)
11. p = pa + dgh, for g (hydrodynamics) 12. 2Q = 2I + A + S, for I (nuclear physics)
mv2
13. Fc =
, for r r
(centripetal force)
4F 14. P = pD2, for F (automotive trades)
15.
ST
=
A 5T
+
0.05d, for A
(welding)
16. u = - 2emL , for L (spectroscopy)
17. ct2 = 0.3t - ac, for a (medical technology)
18. 2p + dv2 = 2d1C - W2, for W (fluid flow)
19. T = c + d, for d (traffic flow) v
20. B = m0I , for R (magnetic field) 2pR
21.
K1 K2
=
m1
+ m1
m2
,
for
m2
(kinetic energy)
F
22.
f=
d
-
, for d F
(photography)
2mg
23.
a
=
M
+
, 2m
for
M
(pulleys)
V1m + M2
24. v =
, for M (ballistics)
m
25.
C
2 0
=
C1211
+
2V2, for V
(electronics)
26. A1 = A1M + 12, for M (photography)
27. N = r1A - s2, for s (engineering)
28. T = 31T2 - T12, for T1 (oil drilling)
29. T2 = T1 - 1h00, for h (air temperature) 30. p2 = p1 + rp111 - p12, for r (population growth)
31. Q1 = P1Q2 - Q12, for Q2 (refrigeration)
32. p - pa = dg1y2 - y12, for y1 (fire science)
33. N = N1T - N211 - T2, for N1 (machine design)
34. ta = tc + 11 - h2tm, for h (computer access time)
35. L = p1r1 + r22 + 2x1 + x2, for r1 (pulleys)
36.
I
=
VR2
+
VR111
+
m2 , for m
(electronics)
R1R2
37.
P
=
V11V2 gJ
V12 , for V2
(jet engine power)
38. W = T1S1 - S22 - Q, for S2 (refrigeration)
39.
C
=
d
2eAk1k2 1k1 + k22
,
for
e
(electronics)
3LPx2 - Px3
40. d =
, for L (beam deflection)
6EI
1.11 Formulas and Literal Equations
45
41.
V
=
Ca1
-
n b, for n N
(property deprecation)
42.
p P
=
B
AI +
, AI
for
B
(atomic theory)
In Exercises 43–48, find the indicated values.
43. For a car’s cooling system, the equation p1C - n2 + n = A is used. If p = 0.25, C = 15.0 L, and A = 13.0 L, solve for n
(in L).
44. A formula used in determining the total transmitted power Pt in an AM radio signal is Pt = Pc11 + 0.500m22. Find Pc if Pt = 680 W and m = 0.925.
45. A formula relating the Fahrenheit temperature F and the Celsius
temperature C is F
=
9 5
C
+
32. Find the Celsius temperature that
corresponds to 90.2°F.
46. In forestry, a formula used to determine the volume V of a log is
V
=
1 2
L
1B
+
b2, where L is the length of the log and B and
b are
the areas of the ends. Find b (in ft2) if V = 38.6 ft3, L = 16.1 ft,
and B = 2.63 ft2. See Fig. 1.17.
L
b
B
Fig. 1.17
47. The voltage V1 across resistance I
R1 is V1 = R1V+R1R2, where V is
the voltage across resistances R1
and R2. See Fig. 1.18. Find R2 (in
V
Ω) if R1 = 3.56 Ω, V1 = 6.30 V,
and V = 12.0 V.
R1
V1
R2
48. The efficiency E of a computer
multiprocessor compilation is
given
by
E
=
q
+
1 p11
-
q2 ,
Fig. 1.18
where p is the number of processors and q is the fraction of the
compilation that can be performed by the available parallel proces-
sors. Find p for E = 0.66 and q = 0.83.
In Exercises 49 and 50, set up the required formula and solve for the indicated letter.
49. One missile travels at a speed of v2 mi/h for 4 h, and another missile travels at a speed of v1 for t + 2 hours. If they travel a total of d mi, solve the resulting formula for t.
50. A microwave transmitter can handle x telephone communications, and 15 separate cables can handle y connections each. If the combined system can handle C connections, solve for y.
answers to Practice Exercises
1. u - kA
2. P + nc n
46
ChaPTER 1 Basic Algebraic Operations
1.12 Applied Word Problems
Procedure for Solving word Problems • Identifying the Unknown quantities • Setting Up the Proper Equation • Examples of Solving word Problems
Many applied problems are at first word problems, and we must put them into mathematical terms for solution. Usually, the most difficult part in solving a word problem is identifying the information needed for setting up the equation that leads to the solution. To do this, you must read the problem carefully to be sure that you understand all of the terms and expressions used. Following is an approach you should use.
■ See Appendix A, page A-1, for a variation to the method outlined in these steps. You
might find it helpful.
Procedure for Solving word Problems
1. Read the statement of the problem. First, read it quickly for a general overview. Then reread slowly and carefully, listing the information given.
2. Clearly identify the unknown quantities and then assign an appropriate letter to represent one of them, stating this choice clearly.
3. Specify the other unknown quantities in terms of the one in step 2.
4. If possible, make a sketch using the known and unknown quantities.
5. Analyze the statement of the problem and write the necessary equation. This is often the most difficult step because some of the information may be implied and not explicitly stated. Again, a very careful reading of the statement is necessary.
6. Solve the equation, clearly stating the solution.
7. Check the solution with the original statement of the problem.
Read the following examples very carefully and note just how the outlined procedure is followed.
■ Be sure to carefully identify your choice for the unknown. In most problems, there is really a choice. Using the word let clearly shows that a specific choice has been made.
■ The statement after “let x (or some other appropriate letter) = ” should be clear. It should completely define the chosen unknown.
F
17 lb
F+ 3
Fig. 1.19
CAUTION Always check a verbal problem with the original statement of the problem, not the first equation, because it was derived from the statement. ■
E X A M P L E 1 sum of forces on a beam
A 17-lb beam is supported at each end. The supporting force at one end is 3 lb more than at the other end. Find the forces.
Since the force at each end is required, we write
step 2
let F = the smaller force 1in lb2
as a way of establishing the unknown for the equation. Any appropriate letter could be used, and we could have let it represent the larger force.
Also, since the other force is 3 lb more, we write
step 3
F + 3 = the larger force 1in lb2
step 4
We now draw the sketch in Fig. 1.19.
Since the forces at each end of the beam support the weight of the beam, we have the equation
step 5
F + 1F + 32 = 17
This equation can now be solved: 2F = 14
step 6
F = 7 lb
step 7 Thus, the smaller force is 7 lb, and the larger force is 10 lb. This checks with
the original statement of the problem.
■
1.12 Applied Word Problems
47
E X A M P L E 2 office complex energy-efficient lighting
In designing an office complex, an architect planned to use 34 energy-efficient ceiling lights using a total of 1000 W. Two different types of lights, one using 25 W and the other using 40 W, were to be used. How many of each were planned?
Since we want to find the number of each type of light, we
let x = number of 25 W lights
■ “Let x = 25 W lights” is incomplete. We want to find out how many there are.
Also, since there are 34 lights in all, 34 - x = number of 40 W lights
We also know that the total wattage of all lights is 1000 W. This means
■ See Appendix A, page A-1, for a “sketch” that might be used with this example.
25 W each
number
25x
total wattage of 25 W lights
40 W each
number
+ 40134 - x2 = 1000
total wattage of 40 W lights
25x + 1360 - 40x = 1000 -15x = -360
x = 24
total wattage of all lights
Therefore, there are 24 25-W lights and 10 40-W lights. The total wattage of these lights is 241252 + 101402 = 600 + 400 = 1000. We see that this checks with the statement
of the problem.
■
Practice Exercise
1. Solve the problem in Example 3 by letting y = number of slides with 6 mg.
E X A M P L E 3 number of medical slides
A medical researcher finds that a given sample of an experimental drug can be divided into 4 more slides with 5 mg each than with 6 mg each. How many slides with 5 mg each can be made up?
We are asked to find the number of slides with 5 mg, and therefore we
let x = number of slides with 5 mg
Because the sample may be divided into 4 more slides with 5 mg each than of 6 mg each, we know that
x - 4 = number of slides with 6 mg
Because it is the same sample that is to be divided, the total mass of the drug on each set of slides is the same. This means
5 mg each
number
5x
6 mg
number
each
=
61x - 42
total mass 5-mg slides
total mass 6-mg slides
5x = 6x - 24 -x = -24 or x = 24
Therefore, the sample can be divided into 24 slides with 5 mg each, or 20 slides with
6 mg each. Since the total mass, 120 mg, is the same for each set of slides, the solution
checks with the statement of the problem.
■
48
ChaPTER 1 Basic Algebraic Operations
E X A M P L E 4 Distance traveled—space travel
■ A maneuver similar to the one in Example 4 was used on several servicing missions to the Hubble space telescope from 1999 to 2009.
Shuttle
6000 km
Satellite
A space shuttle maneuvers so that it may “capture” an already orbiting satellite that is
6000 km ahead. If the satellite is moving at 27,000 km/h and the shuttle is moving at
29,500 km/h, how long will it take the shuttle to reach the satellite? (All digits shown are
significant.)
First, we let t = the time for the shuttle to reach the satellite. Then, using the fact that
the shuttle must go 6000 km farther in the same time, we draw the sketch in Fig. 1.20. Next, we use the formula distance = rate * time 1d = rt2. This leads to the following equation and solution.
speed of
shuttle
time
29,500t
=
6000
speed of
satellite
time
+
27,000t
distance traveled by shuttle
distance between at beginning
distance traveled by satellite
2500t = 6000 t = 2.400 h
Fig. 1.20
This means that it will take the shuttle 2.400 h to reach the satellite. In 2.400 h, the shuttle
will travel 70,800 km, and the satellite will travel 64,800 km. We see that the solution
checks with the statement of the problem.
■
E X A M P L E 5 mixture—gasoline and methanol
■ “Let x = methanol” is incomplete. We want to find out the volume (in L) that is to be
added.
A refinery has 7250 L of a gasoline-methanol blend that is 6.00% methanol. How much pure methanol must be added so that the resulting blend is 10.0% methanol?
First, let x = the number of liters of methanol to be added. The total volume of methanol in the final blend is the volume in the original blend plus that which is added. This total volume is to be 10.0% of the final blend. See Fig. 1.21.
6.00%
original volume
10%
final volume
7250 6.00%
+
x 100%
=
7250 + x 10.0%
0.0600172502
+
x
=
methanol in
methanol
original mixture
added
435 + x = 725 + 0.100x
0.900x = 290, x = 322 L
0.10017250 + x2
methanol in final mixture
to be added
Liters of methanol Fig. 1.21
Checking (to three significant digits), there would be 757 L of methanol of a total 7570 L. ■
EXERCISES 1.12
In Exercises 1–4, make the given changes in the indicated examples of this section and then solve the resulting problems.
1. In Example 2, in the first line, change 34 to 31. 2. In Example 3, in the second line, change “4 more slides” to “3 more
slides.” 3. In Example 4, in the second line, change 27,000 km/h to
27,100 km/h. 4. In Example 5, change “pure methanol” to “of a blend with 50.0%
methanol.”
In Exercises 5–34, set up an appropriate equation and solve. Data are accurate to two significant digits unless greater accuracy is given.
5. A certain new car costs $5000 more than the same model new car cost 6 years ago. Together a new model today and 6 years ago cost $64,000. What was the cost of each? (Assume all values are exact.)
6. The flow of one stream into a lake is 1700 ft3/s more than the flow of a second stream. In 1 h, 1.98 * 107 ft3 flow into the lake from the two streams. What is the flow rate of each?
1.12 Applied Word Problems
49
7. Approximately 6.9 million wrecked cars are recycled in two consecutive years. There were 500,000 more recycled the second year than the first year. How many are recycled each year?
8. A business website had twice as many hits on the first day of a promotion as on the second day. If the total number of hits for both days was 495,000, find the number of hits on each day.
9. Petroleum rights to 140 acres of land are leased for $37,000. Part of the land leases for $200 per acre, and the reminder for $300 per acre. How much is leased at each price?
10. A vial contains 2000 mg, which is to be used for two dosages. One patient is to be administered 660 mg more than another. How much should be administered to each?
11. After installing a pollution control device, a car’s exhaust contained the same amount of pollutant after 5.0 h as it had in 3.0 h. Before the installation the exhaust contained 150 ppm/h (parts per million per hour) of the pollutant. By how much did the device reduce the emission?
12. Three meshed spur gears have a total of 107 teeth. If the second gear has 13 more teeth than the first and the third has 15 more teeth than the second, how many teeth does each have?
13. A cell phone subscriber paid x dollars per month for the first 6 months. He then increased his data plan, and his bill increased by $10 per month for the next 4 months. If he paid a total of $890 for the 10-month period, find the amount of his bill before and after the increase.
14. A satellite television subscriber paid x dollars per month for the first year. Her monthly bill increased by $30 per month for the second and third years, and then another $20 for the fourth and fifth years. If the total amount paid for the 5-year period was $7320, find the three different monthly bill amounts.
15. The sum of three electric currents that come together at a point in a circuit is zero. If the second current is twice the first and the third current is 9.2 mA more than the first, what are the currents? (The sign indicates the direction of flow.)
16. A delivery firm uses one fleet of trucks on daily routes of 8 h. A second fleet, with five more trucks than the first, is used on daily routes of 6 h. Budget allotments allow for 198 h of daily delivery time. How many trucks are in each fleet?
17. A natural gas pipeline feeds into three smaller pipelines, each of which is 2.6 km longer than the main pipeline. The total length of the four pipelines is 35.4 km. How long is each section?
18. At 100% efficiency two generators would produce 750 MW of power. At efficiencies of 65% and 75%, they produce 530 MW. At 100% efficiency, what power would each produce?
19. A wholesaler sells three types of GPS systems. A dealer orders twice as many economy systems at $40 each, and 75 more econoplus systems at $80 each, than deluxe systems at $140 each, for $42,000. How many of each were ordered?
20. A person won a state lottery prize of $20,000, from which 25% was deducted for taxes. The remainder was invested, partly for a 40% gain, and the rest for a 10% loss. How much was each investment if there was a $2000 net investment gain?
21. Train A is 520 ft long and traveling at 60.0 mi/h. Train B is 440 ft long and traveling at 40 mi/h in the opposite direction of train A on an adjacent track. How long does it take for the trains to completely pass each other? (Footnote: A law was once actually passed by the Wisconsin legislature that included “whenever two trains meet at an intersection . . . , neither shall proceed until the other has.”)
22. A family has $3850 remaining of its monthly income after making the monthly mortgage payment, which is 23.0% of the monthly income. How much is the mortgage payment?
23. A ski lift takes a skier up a slope at 50 m/min. The skier then skis down the slope at 150 m/min. If a round trip takes 24 min, how long is the slope?
24. Before being put out of service, the supersonic jet Concorde made a trip averaging 120 mi/h less than the speed of sound for 1.0 h, and 410 mi/h more than the speed of sound for 3.0 h. If the trip covered 3990 mi, what is the speed of sound?
25. Trains at each end of the 50.0-km-long Eurotunnel under the English Channel start at the same time into the tunnel. Find their speeds if the train from France travels 8.0 km/h faster than the train from England and they pass in 17.0 min. See Fig. 1.22.
England
English Channel
France
17.0 min
50.0 km Fig. 1.22
26. An executive would arrive 10.0 min early for an appointment if traveling at 60.0 mi/h, or 5.0 min early if traveling at 45.0 mi/h. How much time is there until the appointment?
27. One lap at the Indianapolis Speedway is 2.50 mi. In a race, a car stalls and then starts 30.0 s after a second car. The first car travels at 260 ft/s, and the second car travels at 240 ft/s. How long does it take the first car to overtake the second, and which car will be ahead after eight laps?
28. A computer chip manufacturer produces two types of chips. In testing a total of 6100 chips of both types, 0.50% of one type and 0.80% of the other type were defective. If a total of 38 defective chips were found, how many of each type were tested?
29. Two gasoline distributors, A and B, are 228 mi apart on Interstate 80. A charges $2.90/gal and B charges $2.70/gal. Each charges 0.2¢/gal per mile for delivery. Where on Interstate 80 is the cost to the customer the same?
30. An outboard engine uses a gasoline-oil fuel mixture in the ratio of 15 to 1. How much gasoline must be mixed with a gasoline-oil mixture, which is 75% gasoline, to make 8.0 L of the mixture for the outboard engine?
31. A car’s radiator contains 12 L of antifreeze at a 25% concentration. How many liters must be drained and then replaced by pure antifreeze to bring the concentration to 50% (the manufacturer’s “safe” level)?
32. How much sand must be added to 250 lb of a cement mixture that is 22% sand to have a mixture that is 25% sand?
33. To pass a 20-m long semitrailer traveling at 70 km/h in 10 s, how fast must a 5.0-m long car go?
34. An earthquake emits primary waves moving at 8.0 km/s and secondary waves moving at 5.0 km/s. How far from the epicenter of the earthquake is the seismic station if the two waves arrive at the station 2.0 min apart?
answer to Practice Exercise 1. 24 with 5 mg, 20 with 6 mg
50
ChaPTER 1 Basic Algebraic Operations
CHAPTER 1 KEy FORMULAS AND EqUATIONS
Commutative law of addition: a + b = b + a Associative law of addition: a + 1b + c2 = 1a + b2 + c
Commutative law of multiplication: ab = ba Associative law of multiplication: a1bc2 = 1ab2c
Distributive law: a1b + c2 = ab + ac a + 1 -b2 = a - b
am * an = am + n
am an
=
am - n
1m 7 n, a ∙ 02,
1am2n = amn
a0 = 1 1a ∙ 02 2ab = 2a2b 1a and b positive real numbers2
(1.1) a - 1 -b2 = a + b
(1.3)
am
1
an = an - m
1m 6 n, a ∙ 02
(1.5)
1ab2n = anbn,
a n an
ab b
= bn
1b ∙ 02
(1.7)
a-n
=
1 an
1a ∙ 02
(1.9)
(1.2)
(1.4) (1.6) (1.8)
CHAPTER 1 REVIEw EXERCISES
CONCEPT CHECK EXERCISES Determine each of the following as being either true or false. If it is false, explain why.
1. The absolute value of any real number is positive.
2. 16 - 4 , 2 = 14 3. For approximate numbers, 26.7 - 15 = 11.7. 4. 2a3 = 8a3 5. 0.237 = 2.37 * 10-1 6. - 2-4 = 2 7. 4x - 12x + 32 = 2x + 3
8. 1x - 722 = 49 - 14x + x2
9.
6x + 2
2
=
3x
10. If 5x - 4 = 0, x = 5>4
11. If a - bc = d, c = 1d - a2 >b
12. In setting up the solution to a word problem involving numbers of gears, it would be sufficient to “let x = the first gear.”
PRACTICE AND APPLICATIONS In Exercises 13–24, evaluate the given expressions.
13. - 2 + 1 -52 - 3
1 -521621 -42 15. 1 - 22132
17.
-5 -
0 21-62 0
+
- 15 3
14. 6 - 8 - 1 -42
1 - 921 - 1221 - 42
16.
24
18.
3
-
50 -3
-
20
-
0 -40
-4
19.
18 3-5
-
1 -422
21. 216 - 264 23. 1 2722 - 23 8
20.
- 1 -322 -
-8
1-22 - 0 -4 0
22. - 281 + 144 24. - 24 16 + 1 2622
In Exercises 25–32, simplify the given expressions. Where appropriate, express results with positive exponents only.
25. 1 - 2rt222
26. 13a0b-223
27. - 3mn-5t18m-3n42
- 16N -21NT22
29.
- 2N0T -1
15p4q2r 28. 5pq5r
- 35x-1y1x2y2
30.
5xy -1
31. 245
32. 29 + 36
In Exercises 33–36, for each number, (a) determine the number of significant digits and (b) round off each to two significant digits.
33. 8000
34. 21,490
35. 9.050
36. 0.7000
In Exercises 37–40, evaluate the given expressions. All numbers are
approximate.
37. 37.3 - 16.9211.06722
8.896 * 10-12 38. - 3.5954 - 6.0449
20.1958 + 2.844
39.
3.14216522
40.
1 0.03568
+
37,466 29.632
Review Exercises
51
In Exercises 41–46, make the indicated conversions.
41. 875 Btu to joules
42. 18.4 in. to meters
43. 65 km/h to ft/s
44. 12.25 g/L to lb/ft3
45. 225 hp to joules per minute 46. 89.7 lb/in.2 to N/cm2
In Exercises 47–78, perform the indicated operations.
47. a - 3ab - 2a + ab 49. 6LC - 13 - LC2 51. 12x - 1215 + x2
48. xy - y - 5y - 4xy 50. - 12x - b2 - 31 - x - 5b2 52. 1C - 4D21D - 2C2
53. 1x + 822
54. 12r - 9s22
2h3k2 - 6h4k5
55.
2h2k
57. 4R - 32r - 13R - 4r24
4a2x3 - 8ax4
56.
- 2ax2
58. - 3b - 33a - 1a - 3b24 + 4a
59. 2xy - 53z - 35xy - 17z - 6xy246
60. x2 + 3b + 31b - y2 - 312b - y + z2 4
61. 12x + 12 1x2 - x - 32
62. 1x - 32 12x2 + 1 - 3x2
63. - 3y1x - 4y22
64. - s14s - 3t22
65. 3p31q - p2 - 2p11 - 3q24
66. 3x32y - r - 41x - 2r24
12p3q2 - 4p4q + 6pq5
67.
2p4q
27s3t2 - 18s4t + 9s2t
68.
- 9s2t
69. 12x2 + 7x - 302 , 1x + 62 70. 14x2 - 412 , 12x + 72
3x3 - 7x2 + 11x - 3
71.
3x - 1
w3 + 7w - 4w2 - 12
72.
w-3
4x4 + 10x3 + 18x - 1
73.
x+3
8x3 - 14x + 3
74.
2x + 3
75. - 351r + s - t2 - 2313r - 2s2 - 1t - 2s246
76. 11 - 2x21x - 32 - 1x + 4214 - 3x2
2y3 - 7y + 9y2 + 5
77.
2y - 1
6x2 + 5xy - 4y2
78.
2x - y
In Exercises 79–90, solve the given equations.
79. 3x + 1 = x - 8
80. 4y - 3 = 5y + 7
5x 3
81.
7
= 2
83. - 6x + 5 = - 31x - 42
214 - N2 5
82.
-3
= 4
84. - 21 - 4 - y2 = 3y
85. 2s + 413 - s2 = 6
86. 2 0 x 0 - 1 = 3
87. 3t - 217 - t2 = 512t + 12
88. - 18 - x2 = x - 212 - x2
89. 2.7 + 2.012.1x - 3.42 = 0.1
90. 0.25016.721 - 2.44x2 = 2.08
In Exercises 91–100, change numbers in ordinary notation to scientific notation or change numbers in scientific notation to ordinary notation. (See Appendix B for an explanation of the symbols that are used.)
91. A certain computer has 60,000,000,000,000 bytes of memory.
92. The escape velocity (the velocity required to leave the Earth’s gravitational field) is about 25,000 mi/h.
93. In 2015, the most distant known object in the solar system, a dwarf planet named V774104, was discovered. It was 15,400,000,000 km from the sun.
94. Police radar has a frequency of 1.02 * 109 Hz. 95. Among the stars nearest the Earth, Centaurus A is about
2.53 * 1013 mi away. 96. Before its destruction in 2001, the World Trade Center had nearly
107 ft2 of office space.
97. The faintest sound that can be heard has an intensity of about 10-12 W/m2.
98. An optical coating on glass to reduce reflections is about 0.00000015 m thick.
99. The maximum safe level of radiation in the air of a home due to radon gas is 1.5 * 10-1 Bq>L. (Bq is the symbol for bequerel, the metric unit of radioactivity, where 1 Bq = 1 decay>s.)
100. A certain virus was measured to have a diameter of about 0.00000018 m.
In Exercises 101–114, solve for the indicated letter. Where noted, the given formula arises in the technical or scientific area of study.
101. V = pr2L, for L (oil pipeline volume)
2GM 102. R = c2 , for G (astronomy: black holes)
p2EI 103. P = L2 , for E (mechanics)
104. f = p1c - 12 - c1p - 12, for p (thermodynamics)
105. Pp + Qq = Rr, for q (moments of forces)
106. V = IR + Ir, for R (electricity)
107. d = 1n - 12A, for n (optics)
108. mu = 1m + M2v, for M (physics: momentum)
109. N1 = T1N2 - N32 + N3, for N2 (mechanics: gears)
110.
Q
=
kAt1T2 L
T12 , for T1
(solar heating)
111.
R
=
A1T2 H
T12 , for T2
(thermal resistance)
112.
Z2a1
-
lb 2a
=
k, for l
(radar design)
113. d = kx2331a + b2 - x4, for a (mechanics: beams) 114. V = V031 + 3a1T2 - T12 4, for T2 (thermal expansion)
In Exercises 115–120, perform the indicated calculations.
115. A computer’s memory is 5.25 * 1013 bytes, and that of a model 30 years older is 6.4 * 104 bytes. What is the ratio of the newer computer’s memory to the older computer’s memory?
52
ChaPTER 1 Basic Algebraic Operations
116. The time (in s) for an object to fall h feet is given by the expression 0.252h. How long does it take a person to fall 66 ft from a sixth-floor window into a net while escaping a fire?
117. The CN Tower in Toronto is 0.553 km high. The Willis Tower (formerly the Sears Tower) in Chicago is 442 m high. How much higher is the CN Tower than the Willis Tower?
118. The time (in s) it takes a computer to check n cells is found by evaluating 1n>265022. Find the time to check 4.8 * 103 cells.
119. The combined electric resistance of two parallel resistors is found by evaluating the expression R1R1+R2R2. Evaluate this for R1 = 0.0275 Ω and R2 = 0.0590 Ω.
120. The distance (in m) from the Earth for which the gravitational force of the Earth on a spacecraft equals the gravitational force of the sun on it is found by evaluating 1.5 * 10112m>M, where m and M are the masses of the Earth and sun, respectively. Find this distance for m = 5.98 * 1024 kg and M = 1.99 * 1030 kg.
In Exercises 121–124, simplify the given expressions.
121. One transmitter antenna is 1x - 2a2 ft long, and another is 1x + 2a2 yd long. What is the sum, in feet, of their lengths?
122. In finding the value of an annuity, the expression 1Ai - R211 + i22 is used. Multiply out this expression.
123. A computer analysis of the velocity of a link in an industrial robot leads to the expression 41t + h2 - 21t + h22. Simplify this expression.
124. When analyzing the motion of a communications satellite, the
expression
k 2r
-
2h2k k 2r
+
h2rv2
is
used.
Perform
the
indicated
division.
In Exercises 125–136, solve the given problems.
125. Does the value of 3 * 18 , 19 - 62 change if the parentheses are removed?
126. Does the value of 13 * 182 , 9 - 6 change if the parentheses are removed?
127. In solving the equation x - 13 - x2 = 2x - 3, what conclusion can be made?
128. In solving the equation 7 - 12 - x2 = x + 2, what conclusion can be made?
129. For x = 2 and y = -4, evaluate (a) 2 0 x 0 - 2 0 y 0 ; (b) 2 0 x - y 0 . 130. If a 6 0, write the value of 0 a 0 without the absolute value
symbols.
131. If 3 - x 6 0, solve 0 3 - x 0 + 7 = 2x for x. 132. Solve 0 x - 4 0 + 6 = 3x for x. (Be careful!)
133. Show that 1x - y23 = - 1y - x23.
134. Is division associative? That is, is it true (if b ∙ 0, c ∙ 0) that 1a , b2 , c = a , 1b , c2?
135. What is the ratio of 8 * 10-3 to 2 * 104? 136. What is the ratio of 24 + 36 to 24?
In Exercises 137–154, solve the given problems. All data are accurate to two significant digits unless greater accuracy is given.
137. A certain engine produces 250 hp. What is this power in kilowatts (kW)?
138. The pressure gauge for an automobile tire shows a pressure of 32 lb>in.2. What is this pressure in N>m2?
139.
A110ceNrta# imn.
automobile engine Convert this to foot
produces pounds.
a
maximum torque of
140. A typical electric current density in a wire is 1.2 * 106 A>m2 (A is the symbol for ampere). Express this in mA>cm2.
141. Two computer software programs cost $190 together. If one costs $72 more than the other, what is the cost of each?
142. A sponsor pays a total of $9500 to run a commercial on two different TV stations. One station charges $1100 more than the other. What does each charge to run the commercial?
143. Three chemical reactions each produce oxygen. If the first pro-
duces twice that of the second, the third produces twice that of the first, and the combined total is 560 cm3, what volume is pro-
duced by each?
144. In testing the rate at which a polluted stream flows, a boat that travels at 5.5 mi/h in still water took 5.0 h to go downstream between two points, and it took 8.0 h to go upstream between the same two points. What is the rate of flow of the stream?
145. The voltage across a resistor equals the current times the resistance. In a microprocessor circuit, one resistor is 1200 Ω greater than another. The sum of the voltages across them is 12.0 mV. Find the resistances if the current is 2.4 mA in each.
146. An air sample contains 4.0 ppm (parts per million) of two pollutants. The concentration of one is four times the other. What are the concentrations?
147. One road crew constructs 450 m of road bed in 12 h. If another crew works at the same rate, how long will it take them to construct another 250 m of road bed?
148. The fuel for a two-cycle motorboat engine is a mixture of gasoline and oil in the ratio of 15 to 1. How many liters of each are in 6.6 L of mixture?
149. A ship enters Lake Superior from Sault Ste. Marie, moving toward Duluth at 17.4 km/h. Two hours later, a second ship leaves Duluth moving toward Sault Ste. Marie at 21.8 km/h. When will the ships pass, given that Sault Ste. Marie is 634 km from Duluth?
150. A helicopter used in fighting a forest fire travels at 105 mi/h from the fire to a pond and 70 mi/h with water from the pond to the fire. If a round-trip takes 30 min, how long does it take from the pond to the fire? See Fig. 1.23.
105 mi/h
70 mi/h
30 min Fig. 1.23
Practice Test
53
151. One grade of oil has 0.50% of an additive, and a higher grade has 0.75% of the additive. How many liters of each must be used to have 1000 L of a mixture with 0.65% of the additive?
152. Each day a mining company crushes 18,000 Mg of shale-oil rock, some of it 72 L/Mg and the rest 150 L/Mg of oil. How much of each type of rock is needed to produce 120 L/Mg?
153. An architect plans to have 25% of the floor area of a house in ceramic tile. In all but the kitchen and entry, there are 2200 ft2 of floor area, 15% of which is tile. What area can be planned for the kitchen and entry if each has an all-tile floor?
154. A karat equals 1>24 part of gold in an alloy (for example, 9-karat gold is 9>24 gold). How many grams of 9-karat gold must be mixed with 18-karat gold to get 200 g of 14-karat gold?
155. In calculating the simple interest earned by an investment, the equation P = P0 + P0rt is used, where P is the value after an initial principal P0 is invested for t years at interest rate r. Solve for r, and then evaluate r for P = $7625, P0 = $6250, and t = 4.000 years. Write a paragraph or two explaining (a) your method for solving for r, and (b) the calculator steps used to evaluate r, noting the use of parentheses.
CHAPTER 1 PRACTICE TEST
As a study aid, we have included complete solutions for each Practice Test problem at the back of this book.
In Problems 1–5, evaluate the given expressions. In Problems 3 and 5, the numbers are approximate.
1. 29 + 16
1721 -321 -22 2. 1 - 62102
3.372 * 10-3 3. 7.526 * 1012
1 +621 -22 - 31 -12
4.
02 - 50
5.
346.4 - 23.5 287.7
-
0.9443 13.462 10.1092
In Problems 6–12, perform the indicated operations and simplify. When exponents are used, use only positive exponents in the result.
6. 12a0b-2c32-3
7. 12x + 322
8. 3m21am - 2m32
8a3x2 - 4a2x4
9.
- 2ax2
11. 12x - 321x + 72
6x2 - 13x + 7
10.
2x - 1
12. 3x - 34x - 13 - 2x24
13. Solve for y: 5y - 21y - 42 = 7
14. Solve for x: 31x - 32 = x - 12 - 3d2 15. Convert 245 lb/ft3 to kg/L.
16. Express 0.0000036 in scientific notation.
17. List the numbers -3, 0 -4 0 , -p, 22, and 0.3 in numerical order.
18. What fundamental law is illustrated by 315 + 82 = 3152 + 3182?
19. (a) How many significant digits are in the number 3.0450? (b) Round it off to two significant digits.
20. If P dollars is deposited in a bank that compounds interest n times a year, the value of the account after t years is found by evaluating P11 + i/n2nt, where i is the annual interest rate. Find the value of an account for which P = $1000, i = 5%, n = 2, and t = 3 years (values are exact).
21. In finding the illuminance from a light source, the expression 81100 - x22 + x2 is used. Simplify this expression.
22. The equation L = L031 + a1t2 - t12 4 is used when studying thermal expansion. Solve for t2.
23. An alloy weighing 20 lb is 30% copper. How many pounds of another alloy, which is 80% copper, must be added for the final alloy to be 60% copper?
2
Geometry
LEARNING OUTCOMES
After completion of this chapter, the student should be able to:
• Identify perpendicular and parallel lines
• Identify supplementary, complementary, vertical, and corresponding angles
• Determine interior angles and sides of various triangles
• Use the Pythagorean theorem • Identify and analyze different types
of quadrilaterals and polygons • Identify and analyze circles, arcs, and
interior angles of a circle • Calculate area and perimeter
of a geometric shape • Use an approximation method
to estimate an irregular area • Identify and analyze three-dimensional
geometric figures, including volume, surface area, and dimensions
W hen building the pyramids nearly 5000 years ago, and today when using MRI (magnetic resonance imaging) to detect a tumor in a human being, the size and shape of an object are measured.
Because geometry deals with size and shape, the topics and methods of geometry are important in many of the applications of technology.
Many of the methods of measuring geometric objects were known in ancient times, and most of the geometry used in technology has been known for hundreds of years. In about 300 b.c.e., the Greek mathematician Euclid (who lived and taught in Alexandria, Egypt) organized what was known in geometry. He added many new ideas in a 13-volume set of writings known as the Elements. Centuries later it was translated into various languages, and today is second only to the Bible as the most published book in history.
The study of geometry includes the properties and measurements of angles, lines, and surfaces and the basic figures they form. In this chapter, we review the more important methods and formulas for calculating basic geometric measures, such as area and volume. Technical applications are included from areas such as architecture, construction, instrumentation, surveying and civil engineering, mechanical design, and product design of various types, as well as other areas of engineering.
Geometric figures and concepts are also basic to the development of many areas of mathematics, such as graphing and trigonometry. We will start our study of graphs in Chapter 3 and trigonometry in Chapter 4.
▶ in section 2.5, we see how to find an excellent approximation of the area of an irregular geometric figure, such as Lake ontario, one of the great Lakes between the united states and Canada.
54
2.1 Lines and Angles
55
2.1 Lines and Angles
Basic Angles • Parallel Lines and Perpendicular Lines • Supplementary Angles and Complementary Angles • Angles Between Intersecting Lines • segments of Transversals
■ The use of 360 comes from the ancient Babylonians and their number system based on 60 rather than 10, as we use today. However, the specific reason for the choice of 360 is not known. (One theory is that 360 is divisible by many smaller numbers and is close to the number of days in a year.)
It is not possible to define every term we use. In geometry, the meanings of point, line, and plane are accepted without being defined. These terms give us a starting point for the definitions of other useful geometric terms.
The amount of rotation of a ray (or half-line) about its endpoint is called an angle. A ray is that part of a line (the word line means “straight line”) to one side of a fixed point on the line. The fixed point is the vertex of the angle. One complete rotation of a ray is an angle with a measure of 360 degrees, written as 360°. Some special types of angles are as follows:
Name of angle Right angle Straight angle Acute angle Obtuse angle
Measure of angle 90° 180° Between 0° and 90° Between 90° and 180°
E X A M P L E 1 Basic angles
Figure 2.1(a) shows a right angle (marked as ). The vertex of the angle is point B, and the ray is the half-line BA. Figure 2.1(b) shows a straight angle. Figure 2.1(c) shows an acute angle, denoted as ∠E (or ∠DEF or ∠FED). In Fig. 2.1(d), ∠G is an obtuse angle.
C
F
180°
B
A
E
D
G
Fig. 2.1
(a)
(b)
(c)
(d)
■
If two lines intersect such that the angle between them is a right angle, the lines are perpendicular. Lines in the same plane that do not intersect are parallel. These are illustrated in the following example.
E X A M P L E 2 Parallel lines and perpendicular lines
In Fig. 2.2(a), lines AC and DE are perpendicular (which is shown as AC # DE) because they meet in a right angle (again, shown as ) at B.
In Fig. 2.2(b), lines AB and CD are drawn so they do not meet, even if extended.
Therefore, these lines are parallel (which can be shown as AB 7 CD). In Fig. 2.2(c), AB # BC, DC # BC, and AB 7 DC.
D
D
C
A
B
A B
C
C
D
A
B
Fig. 2.2 (a)
E
(b)
(c)
It will be important to recognize perpendicular sides and parallel sides in many of the
geometric figures in later sections.
■
■ Recognizing complementary angles is important in trigonometry.
If the sum of the measures of two angles is 180°, then the angles are called supplementary angles. Each angle is the supplement of the other. If the sum of the measures of two angles is 90°, the angles are called complementary angles. Each is the complement of the other.
56
CHAPTER 2 Geometry
Practice Exercise 1. What is the measure of the complement
of ∠BAC in Fig. 2.3?
Practice Exercise 2. In Fig. 2.7, if ∠2 = 42°, then ∠5 = ?
F
12
A
34
B Transversal
56
C 78
D
E
Fig. 2.7
a
b
c
d
Fig. 2.8
E X A M P L E 3 Supplementary angles and complementary angles
(a) In Fig. 2.3, ∠BAC = 55°, and ∠DAC = 125°. Because 55° + 125° = 180°, ∠BAC and ∠DAC are supplementary angles. ∠BAD is a straight angle.
(b) In Fig. 2.4, we see that ∠POQ is a right angle, or ∠POQ = 90°. Because ∠POR + ∠ROQ = ∠POQ = 90°, ∠POR is the complement of ∠ROQ (or ∠ROQ is the complement of ∠POR).
C Supplementary
angles
Q Complementary
R
angles
125° 55°
D
B
A
O
P
Fig. 2.3
Fig. 2.4
■
It is often necessary to refer to certain special pairs of angles. Two angles that have a common vertex and a side common between them are known as adjacent angles. If two lines cross to form equal angles on opposite sides of the point of intersection, which is the common vertex, these angles are called vertical angles.
E X A M P L E 4 adjacent angles and vertical angles
(a) In Fig. 2.5, ∠BAC and ∠CAD have a common vertex at A and the common side between them. This means that ∠BAC and ∠CAD are adjacent angles.
(b) In Fig. 2.6, lines AB and CD intersect at point O. Here, ∠AOC and ∠BOD are vertical angles, and they are equal. Also, ∠BOC and ∠AOD are vertical angles and are equal.
Adjacent angles
D C
Vertical
C
angles
A
B
O
A
B
D
Fig. 2.5
Fig. 2.6
■
We should also be able to identify the sides of an angle that are adjacent to the angle. In Fig. 2.5, sides AB and AC are adjacent to ∠BAC, and in Fig. 2.6, sides OB and OD are adjacent to ∠BOD. Identifying sides adjacent and opposite an angle in a triangle is important in trigonometry.
In a plane, if a line crosses two or more parallel or nonparallel lines, it is called a
transversal. In Fig. 2.7, AB 7 CD, and the transversal of these two parallel lines is the
line EF. When a transversal crosses a pair of parallel lines, certain pairs of equal angles result.
In Fig. 2.7, the corresponding angles are equal (that is, ∠1 = ∠5, ∠2 = ∠6, ∠3 = ∠7, and ∠4 = ∠8). Also, the alternate-interior angles are equal (∠3 = ∠6 and ∠4 = ∠5), and the alternate-exterior angles are equal (∠1 = ∠8 and ∠2 = ∠7).
When more than two parallel lines are crossed by two transversals, such as is shown in Fig. 2.8, the segments of the transversals between the same two parallel lines are called corresponding segments. A useful theorem is that the ratios of corresponding segments of the transversals are equal. In Fig. 2.8, this means that
ac
=
(2.1)
bd
2.1 Lines and Angles
57
x 6.50 f t
5.65 f t
7.75 f t
Fig. 2.9
E X A M P L E 5 segments of transversals
In Fig. 2.9, part of the beam structure within a building is shown. The vertical beams are parallel. From the distances between beams that are shown, determine the distance x between the middle and right vertical beams.
Using Eq. (2.1), we have
6.50 x =
5.65 7.75
6.5017.752
x=
= 8.92 ft rounded off
■
5.65
EXERCISES 2.1
In Exercises 1–4, answer the given questions about the indicated examples of this section.
1. In Example 2, what is the measure of ∠ABE in Fig. 2.2(a)?
2. In Example 3(b), if ∠POR = 35° in Fig. 2.4, what is the measure of ∠QOR?
3. In Example 4, how many different pairs of adjacent angles are there in Fig. 2.6?
4. In Example 5, if the segments of 6.50 ft and 7.75 ft are interchanged, (a) what is the answer, and (b) is the beam along which x is measured more nearly vertical or more nearly horizontal?
In Exercises 5–12, identify the indicated angles and sides in Fig. 2.10. In Exercises 9 and 10, also find the measures of the indicated angles.
5. Two acute angles
6. Two right angles
7. The straight angle
8. The obtuse angle
9. If ∠CBD = 65°, find its complement.
E
D
10. If ∠CBD = 65°, find its supplement.
11. The sides adjacent to ∠DBC
A
C
B
12. The acute angle adjacent to ∠DBC
Fig. 2.10
In Exercises 13–15, use Fig. 2.11. In Exercises 16–18, use Fig. 2.12. Find the measures of the indicated angles.
13. ∠AOB 16. ∠3
14. ∠AOC 17. ∠4
15. ∠BOD 18. ∠5
F C
50° A
O
E
D B
1= 2
150° 1
4 3
5 2
Fig. 2.11
Fig. 2.12
In Exercises 19–24, find the measures of the angles in Fig. 2.13. 19. ∠1 20. ∠2 21. ∠3 22. ∠4 23. ∠5 24. ∠6
F
AB CD
A
5 62°
1
6
B
AF || BE
D
E
2 C
3
4D
A
44°
C
B
Fig. 2.13
Fig. 2.14
In Exercises 25–30, find the measures of the angles in the truss shown in Fig. 2.14. A truss is a rigid support structure that is used in the construction of buildings and bridges.
25. ∠BDF 28. ∠DBE
26. ∠ABE 29. ∠DFE
27. ∠DEB 30. ∠ADE
In Exercises 31–34, find the indicated distances between the straight irrigation ditches shown in Fig. 2.15. The vertical ditches are parallel.
31. a
32. b
33. c
34. d
3.20 m
3.05 m
c Fig. 2.15
4.75 m a
5.05 m
6.25 m
b d
In Exercises 35–40, find all angles of the given measures for the beam support structure shown in Fig. 2.16.
35. 25° 36. 45° 37. 65° 38. 70° 39. 110° 40. 115°
H I
G F
50°
20°
20°
AB C DE
Fig. 2.16
BCH = DCG
58
CHAPTER 2 Geometry
In Exercises 41–46, solve the given problems
41. A plane was heading in a direction 58° east of directly north. It then turned and began to head in a direction 18° south of directly east. Find the measure of the obtuse angle formed between the two parts of the trip. See Fig. 2.17.
18° ?
58° Fig. 2.17
42. ∠A = 1x + 202° and ∠B = 13x - 22°. Solve for x if (a) ∠A and ∠B are (a) complementary angles; (b) alternate-interior angles.
43. A steam pipe is connected in sections AB, BC, and CD as shown
in Fig. 2.18. Find ∠BCD if AB 7 CD.
E
C
D
A Fig. 2.18
47° B
44. Part of a laser beam striking a surface is reflected and the remainder passes through (see Fig. 2.19). Find the angle between the surface and the part that passes through.
C Laser beam 28°
AB › CO
A
B
O?
Fig. 2.19
45. An electric circuit board has equally spaced parallel wires with connections at points A, B, and C, as shown in Fig. 2.20. How far is A from C, if BC = 2.15 cm?
A
B
Fig. 2.20
C
46. Find the distance on Dundas St. W between Dufferin St. and Ossington Ave. in Toronto, as shown in Fig. 2.21. The north–south streets are parallel.
Bloor St. W
550 m 860 m
St. Clarens Ave. Du erin St.
Ossington Ave.
Dundas St. W
590 m
Fig. 2.21
In Exercises 47–50, solve the given problems related to Fig. 2.22.
47. ∠1 + ∠2 + ∠3 = ? 48. ∠4 + ∠2 + ∠5 = ?
49. Based on Exercise 48, what conclusion can be drawn about a closed geometric figure like the one with vertices at A, B, and D?
Fig. 2.22
50. The angle of elevation is the angle above horizontal that an observer must look to see a higher object. The angle of depression is the angle below horizontal that an observer must look to see a lower object. See Fig. 2.23. Do the angle of elevation and the angle of depression always have the same measure? Explain why or why not.
Angle of depression
Horizontal
Horizontal
Angle of elevation
Fig. 2.23
Answers to Practice Exercises 1. 35° 2. 138°
2.2 Triangles
Types and Properties of Triangles • Perimeter and Area • Hero’s Formula • Pythagorean Theorem • Similar Triangles
■ The properties of the triangle are important in the study of trigonometry, which we start in Chapter 4.
When part of a plane is bounded and closed by straight-line segments, it is called a polygon, and it is named according to the number of sides it has. A triangle has three sides, a quadrilateral has four sides, a pentagon has five sides, a hexagon has six sides, and so on. The most important polygons are the triangle, which we consider in this section, and the quadrilateral, which we study in the next section.
TyPES AND PROPERTIES OF TRIANGLES In a scalene triangle, no two sides are equal in length. In an isosceles triangle, two of the sides are equal in length, and the two base angles (the angles opposite the equal sides) are equal. In an equilateral triangle, the three sides are equal in length, and each of the three angles is 60°.
The most important triangle in technical applications is the right triangle. In a right triangle, one of the angles is a right angle. The side opposite the right angle is the hypotenuse, and the other two sides are called legs.
2.2 Triangles
59
6 cm
5 cm
Fig. 2.24
(a)
E X A M P L E 1 Types of triangles
Figure 2.24(a) shows a scalene triangle; each side is of a different length. Figure 2.24(b) shows an isosceles triangle with two equal sides of 2 in. and equal base angles of 40°. Figure 2.24(c) shows an equilateral triangle with equal sides of 5 cm and equal angles of 60°. Figure 2.24(d) shows a right triangle. The hypotenuse is side AB.
4 cm 2 in.
40°
2 in. 40°
(b)
B
60°
5 cm
5 cm
Hypotenuse
60°
60°
5 cm (c)
C
A
(d)
■
NOTE → [One very important property of a triangle is that the sum of the measures of the three angles of a triangle is 180°.]
In the next example, we show this property by using material from Section 2.1.
E
D
1
3
2
C AB EC
4 A
5 B
Fig. 2.25
E X A M P L E 2 sum of angles of triangle
In Fig. 2.25, because ∠1, ∠2, and ∠3 constitute a straight angle, ∠1 + ∠2 + ∠3 = 180°
Also, by noting alternate interior angles, we see that ∠1 = ∠4 and ∠3 = ∠5. Therefore, by substitution, we have
∠4 + ∠2 + ∠5 = 180°
see Exercises 47–49 of Section 2.1
Therefore, if two of the angles of a triangle are known, the third may be found by sub-
tracting the sum of the first two from 180°.
■
A 30°
150°
W 90° N
Fig. 2.26
? B
Practice Exercise 1. If the triangle in Fig. 2.27 is isosceles and
the vertex angle (at the left) is 30°, what are the base angles?
E X A M P L E 3 sum of angles—airplane flight
An airplane is flying north and then makes a 90° turn to the west. Later, it makes another left turn of 150°. What is the angle of a third left turn that will cause the plane to again fly north? See Fig. 2.26.
From Fig. 2.26, we see that the interior angle of the triangle at A is the supplement of 150°, or 30°. Because the sum of the measures of the interior angles of the triangle is 180°, the interior angle at B is
∠B = 180° - 190° + 30°2 = 60°
The required angle is the supplement of 60°, which is 120°.
■
A line segment drawn from a vertex of a triangle to the midpoint of the opposite side
is called a median of the triangle. A basic property of a triangle is that the three medians
meet at a single point, called the centroid of the triangle. See Fig. 2.27. Also, the three
angle bisectors (lines from the vertices that divide the angles in half) meet at a common
point. See Fig. 2.28.
Angle bisectors
Altitudes
Medians
Centroid
Fig. 2.27
Fig. 2.28
Fig. 2.29
An altitude (or height) of a triangle is the line segment drawn from a vertex perpendicular to the opposite side (or its extension), which is called the base of the triangle. The three altitudes of a triangle meet at a common point. See Fig. 2.29. The three common points of the medians, angle bisectors, and altitudes are generally not the same point for a given triangle.
60
CHAPTER 2 Geometry
2.56 m
3.22 m
4.89 m Fig. 2.30
PERIMETER AND AREA OF A TRIANGLE We now consider two of the most basic measures of a plane geometric figure. The first of these is its perimeter, which is the total distance around it. In the following example, we find the perimeter of a triangle.
E X A M P L E 4 Perimeter of triangle
A roof has triangular trusses with sides of 2.56 m, 3.22 m, and 4.89 m. Find the perimeter of one of these trusses. See Fig. 2.30.
Using the definition, the perimeter of one of these trusses is p = 2.56 + 3.22 + 4.89 = 10.67 m
The perimeter is 10.67 m (to hundredths, since each side is given to hundredths). ■
The second important measure of a geometric figure is its area. Although the concept of area is primarily intuitive, it is easily defined and calculated for the basic geometric figures. Area gives a measure of the surface of the figure, just as perimeter gives the measure of the distance around it.
The area A of a triangle of base b and altitude h is
A
=
1 2
bh
(2.2)
The following example illustrates the use of Eq. (2.2).
5.75 in.
E X A M P L E 5 area of triangle
Find the areas of the triangles in Fig. 2.31(a) and Fig. 2.31(b). Even though the triangles are of different shapes, the base b of each is 16.2 in., and
the altitude h of each is 5.75 in. Therefore, the area of each triangle is
A
=
1 2
bh
=
1 2
116.22
15.752
=
46.6 in.2
■
16.2 in. (a)
(b) Fig. 2.31
16.2 in.
Another formula for the area of a triangle that is particularly useful when we have a
triangle with three known sides and no right angle is Hero’s formula, which is
■ Named for Hero (or Heron), a first-century Greek mathematician.
A = 2s1s - a21s - b21s - c2,
where s
=
1 2
1a
+
b
+
c2
(2.3)
In Eq. (2.3), a, b, and c are the lengths of the sides, and s is one-half of the perimeter.
Fig. 2.32
187 f t 293 f t
206 f t
E X A M P L E 6 Hero’s formula—area of land parcel
A surveyor measures the sides of a triangular parcel of land between two intersecting straight roads to be 206 ft, 293 ft, and 187 ft. Find the area of this parcel (see Fig. 2.32).
To use Eq. (2.3), we first find s:
s
=
1 2
1206
+
293
+
1872
=
1 2
16862
=
343 ft
Now, substituting in Eq. (2.3), we have
A = 23431343 - 20621343 - 29321343 - 1872 = 19,100 ft2
Fig. 2.33
The calculator solution is shown in Fig. 2.33. Note that the value of s was stored in
memory (as x) using the STO▶ key and then used to find A. Using the calculator, it is
not necessary to write down anything but the final result, properly rounded off.
■
2.2 Triangles
61
■ Named for the Greek mathematician Pythagoras (sixth century B.C.E.)
c
a
b Fig. 2.34
THE PyTHAGOREAN THEOREM As we have noted, one of the most important geometric figures in technical applications is the right triangle. A very important property of a right triangle is given by the Pythagorean theorem, which states that
in a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.
If c is the length of the hypotenuse and a and b are the lengths of the other two sides (see Fig. 2.34), the Pythagorean theorem is
CAUTION The Pythagorean theorem applies only to right triangles. ■
c2 = a2 + b2
(2.4)
Pole C
Guy wire
3.20 m
2.65 m
B
A
Fig. 2.35
E X A M P L E 7 Pythagorean theorem—length of wire
A pole is perpendicular to the level ground around it. A guy wire is attached 3.20 m up the pole and at a point on the ground, 2.65 m from the pole. How long is the guy wire?
We sketch the pole and guy wire as shown in Fig. 2.35. Using the Pythagorean theorem, and then substituting, we have
AC2 = AB2 + BC2 = 2.652 + 3.202
AC = 22.652 + 3.202 = 4.15 m
The guy wire is 4.15 m long.
■
SIMILAR TRIANGLES
The perimeter and area of a triangle are measures of its size. We now consider the shape of triangles.
Two triangles are similar if they have the same shape (but not necessarily the same size). There are two very important properties of similar triangles.
Properties of Similar Triangles 1. The corresponding angles of similar triangles are equal. 2. The corresponding sides of similar triangles are proportional.
A
C
B
A'
Fig. 2.36
If one property is true, then the other is also true, and therefore, the triangles are similar. In two similar triangles, the corresponding sides are the sides, one in each triangle, that are between the same pair of equal corresponding angles.
E X A M P L E 8 similar triangles
In Fig. 2.36, a pair of similar triangles are shown. They are similar even though the corresponding parts are not in the same position relative to the page. Using standard symbols, we can write ∆ABC ∼ ∆A′B′C′, where ∆ means “triangle” and ∼ means “is similar to.”
The pairs of corresponding angles are A and A′, B and B′, and C and C′. This means B' A = A′, B = B′, and C = C′.
The pairs of corresponding sides are AB and A′B′, BC and B′C′, and AC and A′C′. In order to show that these corresponding sides are proportional, we write
C'
AB
BC
AC d sides of ∆ABC
=
=
A′B′ B′C′ A′C′
d sides of ∆A′B′C′
■
62
CHAPTER 2 Geometry
h
4.0 f t
24.0 f t
3.0 f t
Fig. 2.37
Practice Exercise 2. In Fig. 2.37, knowing the value of h,
what is the distance between the top of the silo and the end of its shadow?
If we know that two triangles are similar, we can use the two basic properties of similar triangles to find the unknown parts of one triangle from the known parts of the other triangle. The next example illustrates this in a practical application.
E X A M P L E 9 similar triangles—height of silo
On level ground, a silo casts a shadow 24 ft long. At the same time, a nearby vertical pole 4.0 ft high casts a shadow 3.0 ft long. How tall is the silo? See Fig. 2.37.
The rays of the sun are essentially parallel. The two triangles in Fig. 2.37 are similar since each has a right angle and the angles at the tops are equal. The other angles must be equal since the sum of the angles is 180°. The lengths of the hypotenuses are of no importance in this problem, so we use only the other sides in stating the ratios of corresponding sides. Denoting the height of the silo as h, we have
h 4.0
=
32.40,
h = 32 ft
We conclude that the silo is 32 ft high.
■
One of the most practical uses of similar geometric figures is that of scale drawings. Maps, charts, architectural blueprints, engineering sketches, and most drawings that appear in books (including many that have already appeared, and will appear, in this text) are familiar examples of scale drawings.
In any scale drawing, all distances are drawn at a certain ratio of the distances that they represent, and all angles are drawn equal to the angles they represent. Note the distances and angles shown in Fig. 2.38 in the following example.
3.5 cm
Toronto
Chicago
Fig. 2.38
E X A M P L E 1 0 scale drawing
In drawing a map of the area shown in Fig. 2.38, a scale of 1 cm = 200 km is used. In measuring the distance between Chicago and Toronto on the map, we find it to be 3.5 cm. The actual distance x between Chicago and Toronto is found from the proportion
scale
T
actual distance S x
200 km
distance on map S
= 3.5 cm
1 cm
or
x = 700 km
2.7 cm Philadelphia
If we did not have the scale but knew that the distance between Chicago and Toronto is 700 km, then by measuring distances on the map between Chicago and Toronto (3.5 cm) and between Toronto and Philadelphia (2.7 cm), we could find the distance between Toronto and Philadelphia. It is found from the following proportion, determined by use of similar triangles:
Practice Exercise 3. On the same map, if the map distance
from Toronto to Boston is 3.3 cm, what is the actual distance?
700 km
y
= 3.5 cm 2.7 cm
2.717002
y = 3.5 = 540 km
■
■ If the result is required to be in miles, we have the following change of units (see Section 1.4).
540 km a 0.6214 mi b = 340 mi 1 km
Similarity requires equal angles and proportional sides. If the corresponding angles and the corresponding sides of two triangles are equal, the two triangles are congruent. As a result of this definition, the areas and perimeters of congruent triangles are also equal. Informally, we can say that similar triangles have the same shape, whereas congruent triangles have the same shape and same size.
2.2 Triangles
63
Congruent
Similar
2 in. 4 in.
2 in. 4 in.
Fig. 2.39
10 in.
E X A M P L E 1 1 similar and congruent triangles
A right triangle with legs of 2 in. and 4 in. is congruent to
any other right triangle with legs of 2 in. and 4 in. It is also
5 in. similar to any right triangle with legs of 5 in. and 10 in., or
any other right triangle that has legs in the ratio of 1 to 2,
since the corresponding sides are proportional. See
Fig. 2.39.
■
EXERCISES 2.2
In Exercises 1–4, answer the given questions about the indicated examples of this section.
1. In Example 2, if ∠1 = 70° and ∠5 = 45° in Fig. 2.25, what is the measure of ∠2?
2. In Example 5, change 16.2 in. to 61.2 in. What is the answer? 3. In Example 7, change 2.65 m to 6.25 m. What is the answer? 4. In Example 9, interchange 4.0 ft and 3.0 ft. What is the answer?
In Exercises 5–8, determine ∠A in the indicated figures.
5. Fig. 2.40 (a)
B
C
6. Fig. 2.40 (b)
48°
7. Fig. 2.40 (c)
84°
8. Fig. 2.40 (d)
A
A
40°
BC
A
(a)
(b)
4
4
C
66°
3 110° 3
B
CA
B
(c)
(d)
Fig. 2.40
In Exercises 9–16, find the area of each triangle.
9. Fig. 2.41(a) 11. Fig. 2.41(c)
10. Fig. 2.41(b) 12. Fig. 2.41(d)
2.2 f t
6.3 f t (a)
9.62 mm 16.0 mm
(b)
239 cm
322 cm
0.862 in.
415 cm (c)
0.535 in. (d)
Fig. 2.41
13. Right triangle with legs 3.46 ft and 2.55 ft 14. Right triangle with legs 234 mm and 342 mm
0.684 in.
15. Isosceles triangle, equal sides of 0.986 m, third side of 0.884 m 16. Equilateral triangle of sides 3200 yd
In Exercises 17–20, find the perimeter of each triangle.
17. Fig. 2.41(c)
18. Fig. 2.41(d)
19. An equilateral triangle of sides 21.5 cm
20. Isosceles triangle, equal sides of 2.45 in., third side of 3.22 in.
In Exercises 21–26, find the third side of the right triangle shown in Fig. 2.42 for the given values. The values in Exercises 21 and 22 are exact.
21. a = 3 in., b = 4 in. 22. a = 5 yd, c = 13 yd 23. a = 13.8 ft, b = 22.7 ft 24. a = 2.48 m, b = 1.45 m 25. a = 175 cm, c = 551 cm 26. b = 0.474 in., c = 0.836 in.
c
a
b Fig. 2.42
In Exercises 27–30, use the right triangle in Fig. 2.43.
27. Find ∠B.
B
28. Find side c. 29. Find the perimeter. 30. Find the area.
c
23° 90.5 cm
38.4 cm Fig. 2.43
In Exercises 31–58, solve the given problems.
31. What is the angle between the bisectors of the acute angles of a right triangle?
32. If the midpoints of the sides of an isosceles triangle are joined, another triangle is formed. What do you conclude about this inner triangle?
33. For what type of triangle is the centroid the same as the intersection of altitudes and the intersection of angle bisectors?
34. Is it possible that the altitudes of a triangle meet, when extended, outside the triangle? Explain.
35. The altitude to the hypotenuse of a right triangle divides the triangle into two smaller triangles. What do you conclude about the original triangle and the two new triangles? Explain.
36. If two triangles have the same three angles, can you conclude that the triangles are congruent? Explain why or why not.
64
CHAPTER 2 Geometry
37. In Fig. 2.44, show that ∆MKL ∼ ∆MNO. 38. In Fig. 2.45, show that ∆ACB ∼ ∆ADC.
L
M K
B
N
D
OC
A
Fig. 2.44
Fig. 2.45
39. In Fig. 2.44, if KN = 15, MN = 9, and MO = 12, find LM.
40. In Fig. 2.45, if AD = 9 and AC = 12, find AB.
41. A perfect triangle is one that has sides that are integers and the perimeter and area are numerically equal integers. Is the triangle with sides 6, 25, and 29 a perfect triangle?
42. Government guidelines require that a sidewalk to street ramp be such that there is no more than 1.0 in. rise for each horizontal 20.0 in. of the ramp. How long should a ramp be for a curb that is 4.0 in. above the street?
43. The angle between the roof sections of an A-frame house is 50°. What is the angle between either roof section and a horizontal rafter?
44. A transmitting tower is supported by a wire that makes an angle of 52° with the level ground. What is the angle between the tower and the wire?
45. An 18.0-ft tall tree is broken in a wind storm such that the top falls and hits the ground 8.0 ft from the base. If the two sections of the tree are still connected at the break, how far up the tree (to the nearest tenth of a foot) was the break?
46. The Bermuda Triangle is sometimes defined as an equilateral triangle 1600 km on a side, with vertices in Bermuda, Puerto Rico, and the Florida coast. Assuming it is flat, what is its approximate area?
47. The sail of a sailboat is in the shape of a right triangle with sides of 8.0 ft, 15 ft, and 17 ft. What is the area of the sail?
48. An observer is 550 m horizontally from the launch pad of a rocket. After the rocket has ascended 750 m, how far is it from the observer?
49. In a practice fire mission, a ladder extended 10.0 ft just reaches the bottom of a 2.50-ft high window if the foot of the ladder is 6.00 ft from the wall. To what length must the ladder be extended to reach the top of the window if the foot of the ladder is 6.00 ft from the wall and cannot be moved?
50. The beach shade shown in
Fig. 2.46 is made up of 30°-
2.00 m
60°-90° triangular sections.
Find x. (In a 30°-60°-90°
triangle, the side opposite
the 30° angle is one-half the
hypotenuse.)
x
Fig. 2.46
51. A rectangular room is 18 ft long, 12 ft wide, and 8.0 ft high. What is the length of the longest diagonal from one corner to another corner of the room?
52. On a blueprint, a hallway is 45.6 cm long. The scale is 1.2 cm = 1.0 m. How long is the hallway?
53. Two parallel guy wires are attached to a vertical pole 4.5 m and 5.4 m above the ground. They are secured on the level ground at points 1.2 m apart. How long are the guy wires?
54. The two sections of a folding door, hinged in the middle, are at right angles. If each section is 2.5 ft wide, how far are the hinges from the far edge of the other section?
55. A 4.0-ft high wall stands 2.0 ft from a building. The ends of a straight pole touch the building and the ground 6.0 ft from the wall. A point on the pole touches the top of the wall. How long is the pole? See Fig. 2.47.
E
Building
Pole Wall
4.0 f t
6.0 f t
2.0 f t
Fig. 2.47
D
A
B
C
Fig. 2.48
56. To find the width ED of a river, a surveyor places markers at A, B, C, and D, as shown in Fig. 2.48. The markers are placed such that
AB 7 ED, BC = 50.0 ft, DC = 312 ft, and AB = 80.0 ft. How
wide is the river?
57. A water pumping station is to be built on a river at point P in order to deliver water to points A and B. See Fig. 2.49. The design requires that ∠APD = ∠BPC so that the total length of piping that will be needed is a minimum. Find this minimum length of pipe.
A
B 6.00 mi C
P 12.0 mi
Fig. 2.49
10.0 mi D
x x - 12 cm
16 cm
Fig. 2.50
58. The cross section of a drainage trough has the shape of an isosceles triangle whose depth is 12 cm less than its width. If the depth is increased by 16 cm and the width remains the same, the area of the cross section is increased by 160 cm2. Find the original depth and width. See Fig. 2.50.
Answers to Practice Exercises 1. 75° 2. 40 ft 3. 660 km
2.3 Quadrilaterals
65
2.3 Quadrilaterals
Types and Properties of quadrilaterals • Perimeter and Area
A quadrilateral is a closed plane figure with four sides, and these four sides form four interior angles. A general quadrilateral is shown in Fig. 2.51.
A diagonal of a polygon is a straight line segment joining any two nonadjacent vertices. The dashed line is one of two diagonals of the quadrilateral in Fig. 2.52.
Fig. 2.51
Fig. 2.52
TyPES OF qUADRILATERALS
A parallelogram is a quadrilateral in which opposite sides are parallel. In a parallelogram, opposite sides are equal and opposite angles are equal. A rhombus is a parallelogram with four equal sides.
A rectangle is a parallelogram in which intersecting sides are perpendicular, which means that all four interior angles are right angles. In a rectangle, the longer side is usually called the length, and the shorter side is called the width. A square is a rectangle with four equal sides.
A trapezoid is a quadrilateral in which two sides are parallel. The parallel sides are called the bases of the trapezoid.
Practice Exercise 1. Develop a formula for the length d of a
diagonal for the rectangle in Fig. 2.53(c).
E X A M P L E 1 Types of quadrilaterals
A parallelogram is shown in Fig. 2.53(a). Opposite sides a are equal in length, as are opposite sides b. A rhombus with equal sides s is shown in Fig. 2.53(b). A rectangle is shown in Fig. 2.53(c). The length is labeled l, and the width is labeled w. A square with equal sides s is shown in Fig. 2.53(d). A trapezoid with bases b1 and b2 is shown in Fig. 2.53(e).
s
s
l
a
b
s b
sw
ws
s
a
s
Fig. 2.53
(a)
(b)
l
s
(c)
(d)
b1
b2
(e)
■
21 in.
36 in.
21 in. Fig. 2.54 Practice Exercise 2. If we did develop perimeter formulas, what is the formula for the perimeter p of a rhombus of side s?
PERIMETER AND AREA OF A qUADRILATERAL The perimeter of a quadrilateral is the sum of the lengths of the four sides.
E X A M P L E 2 Perimeter—window molding
An architect designs a room with a rectangular window 36 in. high and 21 in. wide, with another window above in the shape of an equilateral triangle, 21 in. on a side. See Fig. 2.54. How much molding is needed for these windows?
The length of molding is the sum of the perimeters of the windows. For the rectangular window, the opposite sides are equal, which means the perimeter is twice the length l plus twice the width w. For the equilateral triangle, the perimeter is three times the side s. Therefore, the length L of molding is
L = 2l + 2w + 3s = 21362 + 21212 + 31212
= 177 in.
■
We could write down formulas for the perimeters of the different kinds of triangles and quadrilaterals. However, if we remember the meaning of perimeter as being the total distance around a geometric figure, such formulas are not necessary.
66
CHAPTER 2 Geometry
For the areas of the square, rectangle, parallelogram, and trapezoid, we have the following formulas.
A = s2
Square of side s (Fig. 2.55)
(2.5)
A = lw
Rectangle of length l and width w (Fig. 2.56)
(2.6)
A = bh
Parallelogram of base b and height h (Fig. 2.57)
(2.7)
A
=
1 2
h
1b1
+
b22
Trapezoid of bases b1 and b2 and height h (Fig. 2.58)
(2.8)
s Fig. 2.55
l w
Fig. 2.56
h
b Fig. 2.57
b1
h
b2 Fig. 2.58
Because a rectangle, a square, and a rhombus are special types of parallelograms, the area of these figures can be found from Eq. (2.7). The area of a trapezoid is of importance when we find areas of irregular geometric figures in Section 2.5.
72 ft 45 ft
72 f t
35 f t
72 f t
45 f t
45 f t
Fig. 2.59
E X A M P L E 3 area—park design
A city park is designed with lawn areas in the shape of a right triangle, a parallelogram, and a trapezoid, as shown in Fig. 2.59, with walkways between them. Find the area of each section of lawn and the total lawn area.
A1
=
1 2
bh
=
1 2
1722
1452
=
1600 ft2
A2 = bh = 1722 1452 = 3200 ft2
A3
=
1 2
h1b1
+
b22
=
1 2
1452
172
+
352
=
2400 ft2
The total lawn area is about 7200 ft2.
■
■ The computer microprocessor chip was first commercially available in 1971.
w w + 2.0 Fig. 2.60
Practice Exercise 3. What is the area of the chip in Fig. 2.60?
E X A M P L E 4 Perimeter—computer chip dimensions
The length of a rectangular computer chip is 2.0 mm longer than its width. Find the
dimensions of the chip if its perimeter is 26.4 mm.
Because the dimensions, the length and the width, are required, let w = the width of the chip. Because the length is 2.0 mm more than the width, we know that w + 2.0 = the length of the chip. See Fig. 2.60.
Because the perimeter of a rectangle is twice the length plus twice the width, we have
the equation
21w + 2.02 + 2w = 26.4
because the perimeter is given as 26.4 mm. This is the equation we need. Solving this equation, we have
2w + 4.0 + 2w = 26.4 4w = 22.4 w = 5.6 mm and w + 2.0 = 7.6 mm
Therefore, the length is 7.6 mm and the width is 5.6 mm. These values check with the
statements of the original problem.
■
2.3 Quadrilaterals
67
EXERCISES 2.3
In Exercises 1–4, make the given changes in the indicated examples of this section and then solve the given problems.
1. In Example 1, interchange the lengths of b1 and b2 in Fig. 2.53(e). What type of quadrilateral is the resulting figure?
2. In Example 2, change the equilateral triangle of side 21 in. to a square of side 21 in. and then find the length of molding.
3. In Example 3, change the dimension of 45 ft to 55 ft in each figure and then find the area.
4. In Example 4, change 2.0 mm to 3.0 mm, and then solve.
In Exercises 5–12, find the perimeter of each figure.
5. Square: side of 85 m
6. Rhombus: side of 2.46 ft
7. Rectangle: l = 9.200 in., w = 7.420 in.
8. Rectangle: l = 142 cm, w = 126 cm
9. Parallelogram in Fig. 2.61 10. Parallelogram in Fig. 2.62
11. Trapezoid in Fig. 2.63
12. Trapezoid in Fig. 2.64
3.7 m
27.3 in.
2.5 m
2.7 m
12.6 in.
14.2 in.
Fig. 2.61
0.730 f t
0.362 ft 0.298 f t
0.440 f t
272 cm
Fig. 2.62 392 cm
201 cm
223 cm
0.612 f t Fig. 2.63
672 cm Fig. 2.64
In Exercises 13–20, find the area of each figure.
13. Square: s = 6.4 mm
14. Square: s = 15.6 ft
15. Rectangle: l = 8.35 in., w = 2.81 in.
16. Rectangle: l = 142 cm, w = 126 cm
17. Parallelogram in Fig. 2.61 18. Parallelogram in Fig. 2.62
19. Trapezoid in Fig. 2.63
20. Trapezoid in Fig. 2.64
In Exercises 21–24, set up a formula for the indicated perimeter or area. (Do not include dashed lines.)
21. The perimeter of the figure in Fig. 2.65 (a parallelogram and a square attached)
22. The perimeter of the figure in Fig. 2.66 (two trapezoids attached)
23. Area of figure in Fig. 2.65
24. Area of figure in Fig. 2.66
b
b h
a a
b a
Fig. 2.65
Fig. 2.66
In Exercises 25–46, solve the given problems.
25. If the angle between adjacent sides of a parallelogram is 90°, what conclusion can you make about the parallelogram?
26. What conclusion can you make about the two triangles formed by the sides and diagonal of a parallelogram? Explain.
27. Find the area of a square whose diagonal is 24.0 cm.
28. In a trapezoid, find the angle between the bisectors of the two angles formed by the bases and one nonparallel side.
29. Noting the quadrilateral in Fig. 2.67, determine the sum of the interior angles of a quadrilateral.
Fig. 2.67
30. The sum S of the measures of the interior angles of a polygon with n sides is S = 1801n - 22. (a) Solve for n. (b) If S = 3600°, how many sides does the polygon have?
31. Express the area A of the large rectangle in Fig. 2.68 formed by the smaller rectangles in two ways. What property of numbers is illustrated by the results?
a
b
b
c
b
a a
Fig. 2.68
Fig. 2.69
32. Express the area of the square in Fig. 2.69 in terms of the smaller rectangles into which it is divided. What algebraic expression is illustrated by the results?
33. Noting how a diagonal of a rhombus divides an interior angle, explain why the automobile jack in Fig. 2.70 is in the shape of a rhombus.
12 mm
16 mm
Fig. 2.70
Fig. 2.71
34. Part of an electric circuit is wired in the configuration of a rhombus and one of its altitudes as shown in Fig. 2.71. What is the length of wire in this part of the circuit?
35. A walkway 3.0 m wide is constructed along the outside edge of a square courtyard. If the perimeter of the courtyard is 320 m, (a) what is the perimeter of the square formed by the outer edge of the walkway? (b) What is the area of the walkway?
36. An architect designs a rectangular window such that the width of the window is 18 in. less than the height. If the perimeter of the window is 180 in., what are its dimensions?
68
CHAPTER 2 Geometry
37. Find the area of the cross section of concrete highway support shown in
10
Fig. 2.72. All measurements are in
feet and are exact.
35
30
12 26 12
Fig. 2.72
50
38. A beam support in a building is in the shape of a parallelogram, as shown in Fig. 2.73. Find the area of the side of the beam shown.
3.93 ft 3.5 f t
16 f t
10 f t
10 f t
12 f t
1.80 ft
28 f t
Fig. 2.73
Fig. 2.74
39. Each of two walls (with rectangular windows) of an A-frame
house has the shape of a trapezoid as shown in Fig. 2.74. If a gallon of paint covers 320 ft2, how much paint is required to paint
these walls? (All data are accurate to two significant digits.)
40. A 1080p high-definition widescreen television screen has 1080 pixels in the vertical direction and 1920 pixels in the horizontal direction. If the screen measures 15.8 in. high and 28.0 in. wide, find the number of pixels per square inch.
41. The ratio of the width to the height of a 43.3 cm (diagonal) laptop computer screen is 1.60. What is the width w and height h of the screen?
42. Six equal trapezoidal sections form a conference table in the shape of a hexagon, with a hexagonal opening in the middle. See Fig. 2.75. From the dimensions shown, find the area of the table top.
30.0 in.
60.0 in. 30.0 in.
30.0 in.
Fig. 2.75
43. A fenced section of a ranch is in the shape of a quadrilateral whose sides are 1.74 km, 1.46 km, 2.27 km, and 1.86 km, the last two sides being perpendicular to each other. Find the area of the section.
44. A rectangular security area is enclosed on one side by a wall, and the other sides are fenced. The length of the wall is twice the width of the area. The total cost of building the wall and fence is $13,200. If the wall costs $50.00>m and the fence costs $5.00>m, find the dimensions of the area.
45. What is the sum of the measures of the interior angles of a quadrilateral? Explain.
46. Find a formula for the area of a rhombus in terms of its diagonals d1 and d2. (See Exercise 33.)
Answers to Practice Exercises 1. d = 2l2 + w2 2. p = 4s 3. 43 mm2
2.4 Circles
Properties of Circles • Tangent and Secant Lines • Circumference and Area • Circular Arcs and Angles • Radian Measure of an angle
Diameter Radius Radius
Center Fig. 2.76
A
Secant Chord Tangent
Fig. 2.77 B
O
Fig. 2.78
The next geometric figure we consider is the circle. All points on a circle are at the same distance from a fixed point, the center of the circle. The distance from the center to a point on the circle is the radius of the circle. The distance between two points on the circle on a line through the center is the diameter. Therefore, the diameter d is twice the radius r, or d = 2r. See Fig. 2.76.
There are also certain types of lines associated with a circle. A chord is a line segment having its endpoints on the circle. A tangent is a line that touches (does not pass through) the circle at one point. A secant is a line that passes through two points of the circle. See Fig. 2.77.
An important property of a tangent is that a tangent is perpendicular to the radius drawn to the point of contact. This is illustrated in the following example.
E X A M P L E 1 Tangent line perpendicular to radius
In Fig. 2.78, O is the center of the circle, and AB is tangent at B. If ∠OAB = 25°, find ∠AOB.
Because the center is O, OB is a radius of the circle. A tangent is perpendicular to a radius at the point of tangency, which means ∠ABO = 90° so that
∠OAB + ∠OBA = 25° + 90° = 115°
Because the sum of the angles of a triangle is 180°, we have
∠AOB = 180° - 115° = 65°
■
2.4 Circles
69
■ The symbol p (the Greek letter pi), which we use as a number, was first used in this
way as a number in the 1700s.
CIRCUMFERENCE AND AREA OF A CIRCLE The perimeter of a circle is called the circumference. The formulas for the circumference and area of a circle are as follows:
c = 2pr A = pr2
Circumference of a circle of radius r Area of a circle of radius r
(2.9) (2.10)
■ On the TI-84, values are stored in memory using up to 14 digits with a two-digit
exponent.
Oil 2.4 km
Tubing
Spill
Fig. 2.79
Here, p equals approximately 3.1416. In using a calculator, p can be entered to a much greater accuracy by using the p key.
E X A M P L E 2 area of circle—oil spill
A circular oil spill has a diameter of 2.4 km. It is to be enclosed within special flexible tubing. What is the area of the spill, and how long must the tubing be? See Fig. 2.79.
Since d = 2r, r = d>2 = 1.2 km. Using Eq. (2.10), the area is
A = pr2 = p11.222 = 4.5 km2
The length of tubing needed is the circumference of the circle. Therefore,
c = 2pr = 2p11.22 note that c = pd
= 7.5 km
rounded off
■
Many applied problems involve a combination of geometric figures. The following example illustrates one such combination.
s
s = 3.25 in. Fig. 2.80
E X A M P L E 3 Perimeter and area—machine part
A machine part is a square of side 3.25 in. with a quarter-circle removed (see Fig. 2.80).
Find the perimeter and the area of the part.
Setting up a formula for the perimeter, we add the two sides of length s to one-fourth
of the circumference of a circle with radius s. For the area, we subtract the area of one-
fourth of a circle from the area of the square. This gives
bottom and left
circular section
quarter square circle
p
=
2s
+
2ps 4
=
2s
+
ps 2
A
=
s2
-
ps2 4
where s is the side of the square and the radius of the circle. Evaluating, we have
Practice Exercises 1. Find the circumference of a circle with a
radius of 20.0 cm. 2. Find the area of the circle in Practice
Exercise 1.
p
=
213.252
+
p13.252 2
=
11.6 in.
A
=
3.252
-
p13.2522 4
=
2.27 in.2
■
CIRCULAR ARCS AND ANGLES
An arc is part of a circle, and an angle formed at the center by two radii is a central angle. The measure of an arc is the same as the central angle between the ends of the radii that define the arc. A sector of a circle is the region bounded by two radii and the arc they intercept. A segment of a circle is the region bounded by a chord and its arc. (There are two possible segments for a given chord. The smaller region is a minor segment, and the larger region is a major segment.) These are illustrated in the following example.
70
CHAPTER 2 Geometry
Arc
A Sector
B
Central angle O
Segment
C Fig. 2.81
Inscribed
angle
A
Intercepted arc
P B
C
Fig. 2.82
Arc length equals radius
r 1 rad r
Fig. 2.84
E X A M P L E 4 sector and segment
In Fig. 2.81, a sector of the circle is between radii OA and OB and arc AB (denoted as A¬B).
If the measure of the central angle at O between the radii is 70°, the measure of A¬B is 70°.
A segment of the circle is the region between chord BC and arc BC 1B¬C2.
■
An inscribed angle of an arc is one for which the endpoints of the arc are points on the sides of the angle and for which the vertex is a point (not an endpoint) of the arc. An important property of a circle is that the measure of an inscribed angle is one-half of its intercepted arc.
E X A M P L E 5 inscribed angle
(a) In the circle shown in Fig. 2.82, ∠ABC is inscribed in A¬BC, and it
R
intercepts A¬C. If A¬C = 60°, then ∠ABC = 30°.
(b) In the circle shown in Fig. 2.83, PQ is a diameter, and ∠PRQ is
O
Q
inscribed in the semicircular P¬RQ. Since P¬SQ = 180°, ∠PRQ = 90°.
From this we conclude that an angle inscribed in a semicircle is a
right angle.
■
S
Fig. 2.83
RADIAN MEASURE OF AN ANGLE
To this point, we have measured all angles in degrees. There is another measure of an angle, the radian, that is defined in terms of an arc of a circle. We will find it of importance when we study trigonometry.
If a central angle of a circle intercepts an arc equal in length to the radius of the circle, the measure of the central angle is defined as 1 radian. See Fig. 2.84. The radius can be marked off along the circumference 2p times (about 6.283 times). Thus, 2p rad = 360° (where rad is the symbol for radian), and the basic relationship between radians and degrees is
p rad = 180°
(2.11)
Practice Exercise 3. Express the angle 85.0° in radian
measure.
E X A M P L E 6 Radian measure of an angle
(a) If we divide each side of Eq. (2.11) by p, we get
1 rad = 57.3°
where the result has been rounded off. (b) To change an angle of 118.2° to radian measure, we have
118.2°
=
118.2°
a
p rad 180°
b
=
2.06 rad
Multiplying 118.2° by p rad>180°, the unit that remains is rad, since degrees “cancel.”
We will review radian measure again when we study trigonometry.
■
EXERCISES 2.4
In Exercises 1–4, answer the given questions about the indicated examples of this section.
1. In Example 1, if ∠AOB = 72° in Fig. 2.78, then what is the measure of ∠OAB?
2. In the first line of Example 2, if “diameter” is changed to “radius,” what are the results?
3. In Example 3, if the machine part is the unshaded part (rather than the shaded part) of Fig. 2.80, what are the results?
4. In Example 5(a), if ∠ABC = 25° in Fig. 2.82, then what is the measure of A¬C?
2.4 Circles
71
In Exercises 5–8, refer to the circle with center at O in Fig. 2.85. Iden-
tify the following.
5. (a) A secant line (b) A tangent line
F E
6. (a) Two chords
(b) An inscribed
angle
A
7. (a) Two perpendicular
lines
(b) An isosceles
triangle
B
O
CD
Fig. 2.85
8. (a) A segment (b) A sector with an acute central angle
In Exercises 9–12, find the circumference of the circle with the given radius or diameter.
9. r = 275 ft 11. d = 23.1 mm
10. r = 0.563 m 12. d = 8.2 in.
In Exercises 13–16, find the area of the circle with the given radius or diameter.
13. r = 0.0952 yd 15. d = 2.33 m
14. r = 45.8 cm 16. d = 1256 ft
In Exercises 17 and 18, find the area of the circle with the given circumference.
17. c = 40.1 cm
18. c = 147 m
In Exercises 19–22, refer to Fig. 2.86, where AB is a diameter, TB is a tangent line at B, and ∠ABC = 65°. Determine the indicated angles.
19. ∠CBT
A
20. ∠BCT
21. ∠CAB
22. ∠BTC
O
C
Fig. 2.86
B
T
In Exercises 23–26, refer to Fig. 2.87. Determine the indicated arcs and
angles.
C
23. B¬C 24. A¬B
25. ∠ABC
80°
60°
A
B
26. ∠ACB
Fig. 2.87
In Exercises 27–30, change the given angles to radian measure.
27. 22.5° 28. 60.0° 29. 125.2° 30. 323.0°
In Exercises 31–34, find a formula for the indicated perimeter or area.
31. The perimeter of the quarter-circle in Fig. 2.88.
32. The perimeter of the figure in Fig. 2.89. A quarter-circle is attached to a triangle. Fig. 2.88 r
33. The perimeter of the segment of the
quarter-circle in Fig. 2.88.
b
34. The area of the figure in
Fig. 2.89.
Fig. 2.89
a
r
In Exercises 35–58, solve the given problems.
35. Describe the location of the midpoints of a set of parallel chords of a circle.
36. The measure of A¬B on a circle of radius r is 45°. What is the length of the arc in terms of r and p?
37. In a circle, a chord connects the ends of two perpendicular radii of 6.00 in. What is the area of the minor segment?
38. In Fig. 2.90, chords AB and DE are parallel. What is the relation between A ∆ABC and ∆CDE? Explain.
B C
D
E
Fig. 2.90
39. Equation (2.9) is c = 2pr. Solve for p, and then use the equation d = 2r, where d is the diameter. State the meaning of the result.
40. In 1897, the Indiana House of Representatives passed unani-
mously a bill that included “the . . . important fact that the ratio of
the diameter and circumference is as five-fourths to four.” Under
this definition, what would be the value of p? What is wrong
with this House bill statement? (The bill also passed the Senate
Committee and would have been enacted into law, except for the
intervention of a Purdue professor.)
A = area
41. In Fig. 2.91, for the quarter-circle of radius r,
find the formula for the segment area A in
h
terms of r.
42. For the segment in Fig. 2.91, find
the segment height h in terms of r.
Fig. 2.91
r
43. A person is in a plane 11.5 km above the shore of the Pacific Ocean. How far from the plane can the person see out on the Pacific? (The radius of Earth is 6378 km.)
44. The CN Tower in Toronto has an observation deck at 346 m above the ground. Assuming ground level and Lake Ontario level are equal, how far can a person see from the deck? (The radius of Earth is 6378 km.)
45. An FM radio station emits a signal that is clear within 85 km of the transmitting tower. Can a clear signal be received at a home 68 km west and 58 km south of the tower?
46. A circular pool 12.0 m in diameter has a sitting ledge 0.60 m wide around it. What is the area of the ledge?
47. The radius of the Earth’s equator is 3960 mi. What is the circumference?
48. As a ball bearing rolls along a straight track, it makes 11.0 revolutions while traveling a distance of 109 mm. Find its radius.
72
CHAPTER 2 Geometry
49. The rim on a basketball hoop has an inside diameter of 18.0 in. The largest cross section of a basketball has a diameter of 12.0 in. What is the ratio of the cross-sectional area of the basketball to the area of the hoop?
50. With no change in the speed of flow, by what factor should the diameter of a fire hose be increased in order to double the amount of water that flows through the fire hose?
51. Using a tape measure, the circumference of a tree is found to be 112 in. What is the diameter of the tree (assuming a circular cross section)?
52. Suppose that a 5250-lb force is applied to a hollow steel cylindrical beam that has the cross section shown in Fig. 2.92. The stress on the beam is found by dividing the force by the cross-sectional area. Find the stress.
56. Two pipes, each with a 25.0-mm-diameter hole, lead into a single larger pipe (see Fig. 2.96). In order to ensure proper flow, the cross-sectional area of the hole of the larger pipe is designed to be equal to the sum of the cross-sectional areas of the two smaller pipes. Find the inside diameter of the larger pipe.
Fig. 2.96
2.25 in. 4.00 in.
Fig. 2.92
Fig. 2.93
53. The cross section of a large circular conduit has seven smaller equal circular conduits within it. The conduits are tangent to each other as shown in Fig. 2.93. What fraction of the large conduit is occupied by the seven smaller conduits?
54. Find the area of the room in the plan shown in Fig. 2.94.
57. The velocity of an object moving in a circular path is directed tangent to the circle in which it is moving. A stone on a string moves in a vertical circle, and the string breaks after 5.5 revolutions. If the string was initially in a vertical position, in what direction does the stone move after the string breaks? Explain.
58. Part of a circular gear with 24 teeth is shown in Fig. 2.97. Find the indicated angle.
20° x
9.0 ft
24 ft
35 f t
Fig. 2.94
Fig. 2.95
55. Find the length of the pulley belt shown in Fig. 2.95 if the belt crosses at right angles. The radius of each pulley wheel is 5.50 in.
Fig. 2.97
Answers to Practice Exercises 1. 126 cm 2. 1260 cm2 3. 1.48 rad
2.5 Measurement of Irregular Areas
Trapezoidal Rule • Simpson’s Rule
In practice it may be necessary to find the area of a figure with an irregular perimeter or one for which there is no specific formula. We now show two methods of finding a good approximation of such an area. These methods are particularly useful in technical areas such as surveying, architecture, and mechanical design.
y0 h y1
y2
Fig. 2.98
THE TRAPEZOIDAL RULE
For the area in Fig. 2.98, we draw parallel lines at n equal intervals between the
h yn-2 yn-1
yn
edges to form adjacent trapezoids. The sum of the areas of these trapezoids, all of equal height h, is a good approximation of the area. Now, labeling the lengths of
the parallel lines y0, y1, y2, c , yn, the total area of all trapezoids is
first trapezoid
second trapezoid
third trapezoid
next-to-last trapezoid
last trapezoid
A=
h 2 1y0
+
y12
+
h 2 1y1
+
y22
+
h 2 1y2
+
y32
+
g
+
h 2 1yn - 2
+
yn - 12
+
h 2 1yn - 1
+
yn2
=
h 2
1y0
+
y1
+
y1
+
y2
+
y2
+
y3
+
g
+ yn - 2 + yn - 1 + yn - 1 + yn2
2.5 Measurement of Irregular Areas
73
Therefore, the approximate area is
■ Note carefully that the values of y0 and yn are not multiplied by 2.
A
=
h 2
1y0
+
2y1
+
2y2
+
g
+ 2yn - 1 + yn2
(2.12)
which is known as the trapezoidal rule. The following examples illustrate its use.
2.56 cm 3.82 cm 3.25 cm
2.95 cm 1.85 cm 0.00 cm
2.00 cm Fig. 2.99
E X A M P L E 1 Trapezoidal rule—area of cam
A plate cam for opening and closing a valve is shown in Fig. 2.99. Widths of the face of the cam are shown at 2.00-cm intervals from one end of the cam. Find the area of the face of the cam.
From the figure, we see that
y0 = 2.56 cm y1 = 3.82 cm y2 = 3.25 cm
y3 = 2.95 cm y4 = 1.85 cm y5 = 0.00 cm
(In making such measurements, often a y-value at one end—or both ends—is zero. In such a case, the end “trapezoid” is actually a triangle.) From the given information in this example, h = 2.00 cm. Therefore, using the trapezoidal rule, Eq. (2.12), we have
A
=
2.00 32.56 2
+
213.822
+
213.252
+
212.952
+
211.852
+
0.004
= 26.3 cm2
The area of the face of the cam is approximately 26.3 cm2.
■
When approximating the area with trapezoids, we omit small parts of the area for some trapezoids and include small extra areas for other trapezoids. The omitted areas tend to compensate for the extra areas, which makes the approximation fairly accurate. Also, the use of smaller intervals improves the approximation because the total omitted area or total extra area is smaller.
■ See the chapter introduction.
E X A M P L E 2 Trapezoidal rule—Lake ontario area
From a satellite photograph of Lake Ontario (as shown on page 54), one of the Great Lakes between the United States and Canada, measurements of the width of the lake were made along its length, starting at the west end, at 26.0-km intervals. The widths are shown in Fig. 2.100 and are given in the following table.
Distance from West End (km)
0.0 26.0 52.0 78.0 104 130 156
Toronto
Lake Ontario
Width (km)
Distance from West End (km)
0.0 46.7 52.1 59.2 60.4 65.7 73.9 182 208 234 260 286 312
Niagara Falls
Rochester
Width (km)
87.0 75.5 66.4 86.1 77.0 0.0
Fig. 2.100
Here, we see that y0 = 0.0 km, y1 = 46.7 km, y2 = 52.1 km, . . . , and yn = 0.0 km. Therefore, using the trapezoidal rule, the approximate area of Lake Ontario is found as
follows:
Practice Exercise 1. In Example 2, use only the distances from
west end of (in km) 0.0, 52.0, 104, 156, 208, 260, and 312. Calculate the area and compare with the answer in Example 2.
A
=
26.0 2 30.0
+
2146.72
+
2152.12
+
2159.22
+
2160.42
+
2165.72
+
2173.92
+ 2187.02 + 2175.52 + 2166.42 + 2186.12 + 2177.02 + 0.04
= 19,500 km2
The area of Lake Ontario is actually 19,477 km2.
■
74
CHAPTER 2 Geometry
Parabola Fig. 2.101
h y0
y1
y2
yn- 2
yn - 1
yn h
Fig. 2.102
■ Named for the English mathematician Thomas Simpson (1710–1761).
SIMPSON’S RULE
For the second method of measuring an irregular area, we also draw parallel lines at equal intervals between the edges of the area. We then join the ends of these parallel lines with curved arcs. This takes into account the fact that the perimeters of most figures are curved. The arcs used in this method are not arcs of a circle, but arcs of a parabola. A parabola is shown in Fig. 2.101 and is discussed in detail in Chapter 21. [Examples of parabolas are (1) the path of a ball that has been thrown and (2) the cross section of a microwave “dish.”]
The development of this method requires advanced mathematics. Therefore, we will simply state the formula to be used. It might be noted that the form of the equation is similar to that of the trapezoidal rule.
The approximate area of the geometric figure shown in Fig. 2.102 is given by
A
=
h 3
1y0
+
4y1
+
2y2
+
4y3
+
g
+ 2yn - 2 + 4yn - 1 + yn2
(2.13)
Equation (2.13) is known as Simpson’s rule. CAUTION In using Simpson’s rule, the number n of intervals of width h must be even. ■
E X A M P L E 3 Simpson’s rule—parking lot area
A parking lot is proposed for a riverfront area in a town. The town engineer measured
the widths of the area at 100-ft (three sig. digits) intervals, as shown in Fig. 2.103. Find
the area available for parking.
River
First, we see that there are six intervals, which means Eq. (2.13) may be used. With
y0 = 407 ft, y1 = 483 ft, . . . , y6 = 495 ft, and h = 100 ft, we have
407 f t 483 f t
382 f t 378 f t
285 f t 384 f t 495 f t
Street Fig. 2.103
A
=
100 3 3407
+
414832
+
213822
+
413782
+
212852
+
413842
+
4954
= 241,000 ft2
■
For most areas, Simpson’s rule gives a somewhat better approximation than the trapezoidal rule. The accuracy of Simpson’s rule is also usually improved by using smaller intervals.
E X A M P L E 4 Simpson’s rule—Easter Island area
From an aerial photograph, a cartographer determines the widths of Easter Island (in the Pacific Ocean) at 1.50-km intervals as shown in Fig. 2.104. The widths found are as follows:
Distance from South End (km)
Width (km)
0 1.50 3.00 4.50 6.00 7.50 9.00 10.5 12.0 13.5 15.0 0 4.8 5.7 10.5 15.2 18.5 18.8 17.9 11.3 8.8 3.1
y10 y9
Easter Island
Since there are ten intervals, Simpson’s rule may be used. From the table,
we have the following values: y0 = 0, y1 = 4.8, y2 = 5.7, . . . , y9 = 8.8, y10 = 3.1, and h = 1.5. Using Simpson’s rule, the cartographer would approximate the area of Easter Island as follows:
A
=
1.50 30 3
+
414.82
+
215.72
+
4110.52
+
2115.22
+
4118.52
+ 2118.82 + 4117.92 + 2111.32 + 418.82 + 3.14 = 174 km2 ■
y0 = 0
y2 y1
1.50 km
Fig. 2.104
2.5 Measurement of Irregular Areas
75
EXERCISES 2.5
In Exercises 1 and 2, answer the given questions related to the indicated examples of this section.
1. In Example 1, if widths of the face of the same cam were given at 1.00-cm intervals (five more widths would be included), from the methods of this section, what is probably the most accurate way of finding the area? Explain.
2. In Example 4, if you use only the data from the south end of (in km) 0, 3.00, 6.00, 9.00, 12.0, and 15.0, would you choose the trapezoidal rule or Simpson’s rule to calculate the area? Explain. Do not calculate the area for these data.
In Exercises 3 and 4, answer the given questions related to Fig. 2.105.
3. Which should be more accurate for finding the area, the trapezoidal rule or Simpson’s rule? Explain.
4. If the trapezoidal rule is used to find
the area, will the result probably be
too high, about right, or too little? Explain.
Fig. 2.105
In Exercises 5 and 6, answer the given questions related to Fig. 2.106.
5. If the trapezoidal rule was used to find the area of the region in Fig. 2.106, would the answer be approximate or exact? Explain.
6. Explain why Simpson’s rule cannot be used to find the area of the region in Fig. 2.106.
Fig. 2.106 In Exercises 7–18, calculate the indicated areas. All data are accurate to at least two significant digits.
7. The widths of a kidney-shaped swimming pool were measured at 2.0-m intervals, as shown in Fig. 2.107. Calculate the surface area of the pool, using the trapezoidal rule.
11. Using aerial photography, the widths of an area burned by a forest fire were measured at 0.5-mi intervals, as shown in the following table:
Distance (mi) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Width (mi) 0.6 2.2 4.7 3.1 3.6 1.6 2.2 1.5 0.8
Determine the area burned by the fire by using the trapezoidal rule.
12. Find the area burned by the forest fire of Exercise 11, using Simpson’s rule.
13. A cartographer measured the width of Bruce Peninsula in Ontario at 10-mm intervals on a map (scale 10 mm = 23 km), as shown in Fig. 2.109. The widths are shown in the following list. What is the area of Bruce Peninsula?
Fathom Five National
Marine Park
Tobermory Bruce Peninsula National Park
y0
6
Miller Lake 6
Stokes Bay Lion’s Head
Northern Bruce Peninsula
George Bay
Mar
Colpoy’s
South Bruce Bay
Peninsula
Wiarton
6
Sauble Beach Hepworth
Leith
y8
Shallow Owen Lake Sound
21
Georgian Blu s
Southampton
Tara
Port Elgin
Chatsworth
Saugeen shores
Arran-Elderslie
Fig. 2.109
y0 = 38 mm y3 = 17 mm y6 = 36 mm
y1 = 24 mm y4 = 34 mm y7 = 34 mm
y2 = 25 mm y5 = 29 mm y8 = 30 mm
14. The widths (in m) of half the central arena in the Colosseum in Rome are shown in the following table, starting at one end and measuring from the middle to one side at 4.0-m intervals. Find the area of the arena by the trapezoidal rule. Hint: Remember to double the distances.
Dist. from middle (m) 0.0 4.0 8.0 12.0 16.0 20.0
Width (m)
55.0 54.8 54.0 53.6 51.2 49.0
0.0 m 6.4 m 7.4 m 7.0 m 6.1 m 5.2 m 5.0 m 5.1 m 0.0 m
2.0 m
Fig. 2.107
8. Calculate the surface area of the swimming pool in Fig. 2.107, using Simpson’s rule.
9. The widths of a cross section of an airplane wing are measured at 1.00-ft intervals, as shown in Fig. 2.108. Calculate the area of the cross section, using Simpson’s rule.
1.05 f t 1.15 ft
1.00 ft 0.62 ft
0.52 f t 0.00 f t
Fig. 2.108
0.75 f t
1.00 f t
10. Calculate the area of the cross section of the airplane wing in Fig. 2.108, using the trapezoidal rule.
Dist. Width
24.0 28.0 32.0 36.0 40.0 44.0 45.8 42.0 37.2 31.1 21.7 0.0
15. The widths of the baseball playing area in Boston’s Fenway Park at 45-ft intervals are shown in Fig. 2.110. Find the playing area using the trapezoidal rule.
16. Find the playing area of Fenway Park (see Exercise 15) by Simpson’s rule.
The Green Monster 230 ft 290 ft 330 ft 350 ft 390 ft 410 ft 420 ft
170 ft 360 ft
Fig. 2.110
76
CHAPTER 2 Geometry
17. Soundings taken across a river channel give the following depths with the corresponding distances from one shore.
Distance (ft) 0 50 100 150 200 250 300 350 400 450 500 Depth (ft) 5 12 17 21 22 25 26 16 10 8 0
In Exercises 19–22, calculate the area of the circle by the indicated method.
The lengths of parallel chords of a circle that are 0.250 in. apart are given in the following table. The diameter of the circle is 2.000 in. The distance shown is the distance from one end of a diameter.
Find the area of the cross section of the channel using Simpson’s rule.
18. The widths of a bell crank are measured at 2.0-in. intervals, as shown in Fig. 2.111. Find the area of the bell crank if the two connector holes are each 2.50 in. in diameter.
16.2 in. 18.6 in. 19.0 in. 17.8 in. 12.5 in. 8.2 in.
Fig. 2.111
3.5 in. 6.0 in. 7.6 in. 10.8 in.
2.0 in.
Distance (in.) 0.000 0.250 0.500 0.750 1.000 1.250 1.500 1.750 2.000
Length (in.) 0.000 1.323 1.732 1.936 2.000 1.936 1.732 1.323 0.000
Using the formula A = pr2, the area of the circle is 3.14 in.2.
19. Find the area of the circle using the trapezoidal rule and only the values of distance of 0.000 in., 0.500 in., 1.000 in., 1.500 in., and 2.000 in. with the corresponding values of the chord lengths. Explain why the value found is less than 3.14 in.2.
20. Find the area of the circle using the trapezoidal rule and all values in the table. Explain why the value found is closer to 3.14 in.2 than the value found in Exercise 19.
21. Find the area of the circle using Simpson’s rule and the same table values as in Exercise 19. Explain why the value found is closer to 3.14 in.2 than the value found in Exercise 19.
22. Find the area of the circle using Simpson’s rule and all values in the table. Explain why the value found is closer to 3.14 in.2 than the value found in Exercise 21.
Answer to Practice Exercise 1. 18,100 km2
2.6 Solid Geometric Figures
Rectangular Solid • Cylinder • Prism • Cone • Pyramid • Sphere • Frustum
We now review the formulas for the volume and surface area of some basic solid geometric figures. Just as area is a measure of the surface of a plane geometric figure, volume is a measure of the space occupied by a solid geometric figure.
One of the most common solid figures is the rectangular solid. This figure has six sides (faces), and opposite sides are rectangles. All intersecting sides are perpendicular to each other. The bases of the rectangular solid are the top and bottom faces. A cube is a rectangular solid with all six faces being equal squares.
A right circular cylinder is generated by rotating a rectangle about one of its sides. Each base is a circle, and the cylindrical surface is perpendicular to each of the bases. The height is one side of the rectangle, and the radius of the base is the other side.
A right circular cone is generated by rotating a right triangle about one of its legs. The base is a circle, and the slant height is the hypotenuse of the right triangle. The height is one leg of the right triangle, and the radius of the base is the other leg.
The bases of a right prism are equal and parallel polygons, and the sides are rectangles. The height of a prism is the perpendicular distance between bases. The base of a pyramid is a polygon, and the other faces, the lateral faces, are triangles that meet at a common point, the vertex. A regular pyramid has congruent triangles for its lateral faces.
A sphere is generated by rotating a circle about a diameter. The radius is a line segment joining the center and a point on the sphere. The diameter is a line segment through the center and having its endpoints on the sphere.
The frustum of a cone or pyramid is the solid figure that remains after the top is cut off by a plane parallel to the base.
2.6 Solid Geometric Figures
77
In the following formulas, V represents the volume, A represents the total surface area, S represents the lateral surface area (bases not included), B represents the area of the base, and p represents the perimeter of the base.
h
w l Fig. 2.112
h r
e Fig. 2.113
Rectangular solid (Fig. 2.112)
V = lwh A = 2lw + 2lh + 2wh
Cube (Fig. 2.113)
V = e3 A = 6e2
Right circular cylinder (Fig. 2.114)
V = pr2h A = 2pr2 + 2prh S = 2prh
(2.14) (2.15)
(2.16) (2.17)
(2.18) (2.19) (2.20)
Fig. 2.114
h
Right prism (Fig. 2.115)
V = Bh S = ph
(2.21) (2.22)
s h
r
Fig. 2.115
Right circular cone (Fig. 2.116)
V = 1 pr2h 3
A = pr2 + prs S = prs
(2.23)
(2.24) (2.25)
Fig. 2.116
r Fig. 2.118
s h
Fig. 2.117
r
h
s
R
Fig. 2.119
Regular pyramid (Fig. 2.117)
Sphere (Fig. 2.118)
Frustum of right circular cone (Fig. 2.119)
1 V = Bh
3 1 S = ps 2
V = 4 pr3 3
A = 4pr2
V
=
1 ph1R2 3
+
Rr
+
r22
S = p1R + r2s
(2.26) (2.27)
(2.28) (2.29)
(2.30) (2.31)
Equation (2.21) is valid for any prism, and Eq. (2.26) is valid for any pyramid. There are other types of cylinders and cones, but we restrict our attention to right circular cylinders and right circular cones, and we will often use “cylinder” or “cone” when referring to them.
E X A M P L E 1 Volume of rectangular solid—driveway construction
What volume of concrete is needed for a driveway 25.0 m long, 2.75 m wide, and 0.100 m thick?
The driveway is a rectangular solid for which l = 25.0 m, w = 2.75 m, and h = 0.100 m. Using Eq. (2.14), we have
V = 125.0212.75210.1002
= 6.88 m3
■
78
CHAPTER 2 Geometry
h = 10.4 cm
r = 11.9 cm Fig. 2.120
Practice Exercises 1. Find the volume within the conical cover
in Example 2. 2. Find the surface area (not including the
base) of the storage building in Example 3.
E X A M P L E 2 Total surface area of right circular cone—protective cover
How many square centimeters of sheet metal are required to make a protective coneshaped cover if the radius is 11.9 cm and the height is 10.4 cm? See Fig. 2.120.
To find the total surface area using Eq. (2.24), we need the radius and the slant height s of the cone. Therefore, we must first find s. The radius and height are legs of a right triangle, and the slant height is the hypotenuse. To find s, we use the Pythagorean theorem:
s2 = r2 + h2
Pythagorean theorem
s = 2r2 + h2
solve for s
= 211.92 + 10.42 = 15.8 cm
Now, calculating the total surface area, we have
A = pr2 + prs = p111.922 + p111.92 115.82 = 1040 cm2
Eq. (2.24) substituting
Thus, 1040 cm2 of sheet metal are required.
■
h = 122 f t r = 40.0 f t
Fig. 2.121
E X A M P L E 3 volume of combination of solids—grain storage
A grain storage building is in the shape of a cylinder surmounted by a hemisphere (half a sphere). See Fig. 2.121. Find the volume of grain that can be stored if the height of the
cylinder is 122 ft and its radius is 40.0 ft.
The total volume of the structure is the volume of the cylinder plus the volume of the
hemisphere. By the construction we see that the radius of the hemisphere is the same as the radius of the cylinder. Therefore,
cylinder hemisphere
V
=
pr 2h
+
1 a 4 pr3 b 2 3
=
pr 2h
+
2 pr3 3
= p140.02211222 + 2 p140.023 3
= 747,000 ft3
■
EXERCISES 2.6
In Exercises 1–4, answer the given questions about the indicated examples of this section.
1. In Example 1, if the length is doubled and the thickness is tripled, by what factor is the volume changed?
2. In Example 2, if the value of the slant height s = 17.5 cm is given instead of the height, what is the height?
3. In Example 2, if the radius is halved and the height is doubled, what is the volume?
4. In Example 3, if h is halved, what is the volume?
In Exercises 5–22, find the volume or area of each solid figure for the given values. See Figs. 2.112 to 2.119.
5. Volume of cube: e = 6.95 ft 6. Volume of right circular cylinder: r = 23.5 cm, h = 48.4 cm
7. Total surface area of right circular cylinder: r = 689 mm, h = 233 mm
8. Area of sphere: r = 0.067 in. 9. Volume of sphere: r = 1.037 yd 10. Volume of right circular cone: r = 25.1 m, h = 5.66 m 11. Lateral area of right circular cone: r = 78.0 cm, s = 83.8 cm 12. Lateral area of regular pyramid: p = 345 ft, s = 272 ft 13. Volume of regular pyramid: square base of side 0.76 in.,
h = 1.30 in. 14. Volume of right prism: square base of side 29.0 cm, h = 11.2 cm 15. Volume of frustum of right circular cone:
R = 37.3 mm, r = 28.2 mm, h = 45.1 mm 16. Lateral area of frustum of right circular cone:
R = 3.42 m, r = 2.69 m, s = 3.25 m
2.6 Solid Geometric Figures
79
17. Lateral area of right prism: equilateral triangle base of side 1.092 m, h = 1.025 m
18. Lateral area of right circular cylinder: diameter = 250 ft, h = 347 ft
19. Volume of hemisphere: diameter = 0.65 yd
20. Volume of regular pyramid: square base of side 22.4 m, s = 14.2 m
21. Total surface area of right circular cone: r = 3.39 cm, h = 0.274 cm
22. Total surface area of regular pyramid: All faces and base are equilateral triangles of side 3.67 in. (This is often referred to as a tetrahedron.)
In Exercises 23–46, solve the given problems.
23. Equation (2.28) expresses the volume V of a sphere in terms of the radius r. Express V in terms of the diameter d.
24. Derive a formula for the total surface area A of a hemispherical volume of radius r (curved surface and flat surface).
25. The radius of a cylinder is twice as long as the radius of a cone, and the height of the cylinder is half as long as the height of the cone. What is the ratio of the volume of the cylinder to that of the cone?
26. The base area of a cone is one-fourth of the total area. Find the ratio of the radius to the slant height.
27. In designing a spherical weather balloon, it is decided to double the diameter of the balloon so that it can carry a heavier instrument load. What is the ratio of the final surface area to the original surface area?
28. During a rainfall of 1.00 in., what weight of water falls on an area of 1.00 mi2? Each cubic foot of water weighs 62.4 lb.
29. A rectangular box is to be used to store radioactive materials. The inside of the box is 12.0 in. long, 9.50 in. wide, and 8.75 in. deep. What is the area of sheet lead that must be used to line the inside of the box?
30. A swimming pool is 50.0 ft wide, 78.0 ft long, 3.50 ft deep at one end, and 8.75 ft deep at the other end. How many cubic feet of water can it hold? (The slope on the bottom is constant.) See Fig. 2.122.
78.0 f t
33. A pole supporting a wind turbine is constructed of solid steel and is in the shape of a frustum of a cone. It measures 62.5 m high, and the diameter of the pole at the bottom and top are 3.88 m and 1.90 m, respectively. What is the volume of the pole?
34. A glass prism used in the study of optics has a right triangular base. The legs of the triangle are 3.00 cm and 4.00 cm. The prism is 8.50 cm high. What is the total surface area of the prism? See Fig. 2.124.
4.00 cm
8.50 cm
3.00 cm
Fig. 2.124
35. The Great Pyramid of Egypt has a square base approximately 250 yd on a side. The height of the pyramid is about 160 yd. What is its volume? See Fig. 2.125.
160 yd
Fig. 2.125 36. A paper cup is in the shape of a cone as shown in Fig. 2.126. What
is the surface area of the cup?
3.60 in.
3.50 in.
165 f t
Fig. 2.126
Fig. 2.127
50.0 f t
8.75 f t Fig. 2.122
31. The Alaskan oil pipeline is 750 mi long and has a diameter of 4.0 ft. What is the maximum volume of the pipeline?
32. The volume of a frustum of a pyra-
mid is V
=
1 3
h1a2
+
ab
+
b22
(see
Fig. 2.123). (This equation was dis-
covered by the ancient Egyptians.)
If the base of a statue is the frus-
tum of a pyramid, find its volume
if a = 2.50 m, b = 3.25 m, and
h = 0.750 m.
3.50 f t
b b
h a
a Fig. 2.123
37. Spaceship Earth (shown in Fig. 2.127) at Epcot Center in Florida is a sphere of 165 ft in diameter. What is the volume of Spaceship Earth?
38. A propane tank is constructed in the shape of a cylinder with a hemisphere at each end, as shown in Fig. 2.128. Find the volume of the tank.
6.50 f t
4.00 ft
Fig. 2.128
39. A special wedge in the shape of a regular pyramid has a square base 16.0 mm on a side. The height of the wedge is 40.0 mm. What is the total surface area of the wedge (including the base)?
80
CHAPTER 2 Geometry
40. A lawn roller is a cylinder 0.96.0 m long and 0.60 m in diameter. How many revolutions of the roller are needed to roll 76 m2 of lawn?
41. The circumference of a basketball is about 29.8 in. What is its volume?
42. What is the area of a paper label that is to cover the lateral surface of a cylindrical can 3.00 in. in diameter and 4.25 in. high? The ends of the label will overlap 0.25 in. when the label is placed on the can.
43. The side view of a rivet is shown in Fig. 2.129. It is a conical part on a cylindrical part. Find the volume of the rivet.
0.625 in.
2.75 in.
45. A ball bearing had worn down too much in a machine that was not operating properly. It remained spherical, but had lost 8.0% of its volume. By what percent had the radius decreased?
46. A dipstick is made to measure the volume remaining in the conical container shown in Fig. 2.130. How far below the full mark (at the top of the container) on the stick should the mark for half-full be placed?
18.0 cm
12.0 cm
1.25 in. 0.625 in.
Fig. 2.129 44. A semicircular patio made of concrete 7.5 cm thick has a total
perimeter of 18 m. What is the volume of concrete?
Fig. 2.130
Answers to Practice Exercises 1. 1540 cm3 2. 40,700 ft2
CHAPTER 2 KEy FORMULAS AND EqUATIONS
ac
Line segments
Fig. 2.8
=
(2.1)
bd
Triangle
A
=
1 2
bh
(2.2)
Hero’s formula
A = 2s1s - a21s - b21s - c2,
where s
=
1 2
1a
+
b
+
c2
(2.3)
Pythagorean theorem
Fig. 2.34
c2 = a2 + b2
(2.4)
Square
Fig. 2.55
A = s2
(2.5)
Rectangle
Fig. 2.56
A = lw
(2.6)
Parallelogram Trapezoid Circle
Fig. 2.57 Fig. 2.58
A = bh
A
=
1 2
h1b1
+
b22
c = 2pr
A = pr2
(2.7) (2.8) (2.9) (2.10)
Radians
Fig. 2.84
p rad = 180°
(2.11)
Trapezoidal rule Simpson’s rule
Fig. 2.98 Fig. 2.102
A=
h 2
1y0
+
2y1
+
2y2
+
g
+ 2yn - 1 + yn2
(2.12)
A
=
h 3 1y0
+
4y1
+
2y2
+
4y3
+
g
+ 2yn - 2 + 4yn - 1 + yn2
(2.13)
Review Exercises
81
Rectangular solid Cube Right circular cylinder
Fig. 2.112 Fig. 2.113 Fig. 2.114
Right prism Right circular cone
Fig. 2.115 Fig. 2.116
Regular pyramid
Fig. 2.117
Sphere
Fig. 2.118
Frustum of right circular cone Fig. 2.119
V = lwh A = 2lw + 2lh + 2wh
V = e3 A = 6e2
V = pr2h A = 2pr2 + 2prh S = 2prh
V = Bh S = ph
V
=
1 3
pr
2h
A = pr2 + prs
S = prs
V
=
1 3
Bh
S
=
1 2
ps
V
=
4 3
pr
3
A = 4pr2
V
=
1 3
ph1R2
+
Rr
+
r22
S = p1R + r2s
(2.14) (2.15)
(2.16) (2.17)
(2.18) (2.19) (2.20)
(2.21) (2.22)
(2.23) (2.24) (2.25)
(2.26) (2.27)
(2.28) (2.29)
(2.30) (2.31)
CHAPTER 2 REVIEw EXERCISES
CONCEPT CHECK EXERCISES
Determine each of the following as being either true or false. If it is false, explain why.
1. In Fig. 2.6, ∠AOC is the complement of ∠COB. 2. A triangle of sides 9, 12, and 15 is a right triangle.
3. A quadrilateral with two sides of length a and two sides of length b is always a parallelogram.
4. The circumference of a circle of diameter d is pd.
5. Simpson’s rule could be used to find the approximate area in Example 2 of Section 2.5.
6. The volume of a right circular cylinder is the base area times the height.
PRACTICE AND APPLICATIONS
In Exercises 7–10, use Fig. 2.131. Determine the indicated angles.
7. ∠CGE
IH
8. ∠EGF 9. ∠DGH 10. ∠EGI
C
D
G
AB CD 148°
A
B
Fig. 2.131
E
F
In Exercises 11–18, find the indicated sides of the right triangle shown in Fig. 2.132.
11. a = 12, b = 35, c = ?
12. a = 14, b = 48, c = ?
13. a = 400, b = 580, c = ? 14. b = 5600, c = 6500, a = ?
15. b = 0.380, c = 0.736, a = ?
16. b = 25.1, c = 128, a = ?
a
c
17. b = 38.3, c = 52.9, a = ?
18. a = 0.782, c = 0.885, b = ?
b
Fig. 2.132
In Exercises 19–26, find the perimeter or area of the indicated figure.
19. Perimeter: equilateral triangle of side 8.5 mm 20. Perimeter: rhombus of side 15.2 in. 21. Area: triangle: b = 0.125 ft, h = 0.188 ft 22. Area: triangle of sides 175 cm, 138 cm, 119 cm 23. Circumference of circle: d = 74.8 mm 24. Perimeter: rectangle, l = 2980 yd, w = 1860 yd 25. Area: trapezoid, b1 = 67.2 in., b2 = 126.7 in., h = 34.2 in. 26. Area: circle, d = 0.328 m
82
CHAPTER 2 Geometry
In Exercises 27–32, find the volume of the indicated solid geometric figure.
27. Prism: base is right triangle with legs 26.0 cm and 34.0 cm, height is 14.0 cm
28. Cylinder: base radius 36.0 in., height 2.40 in. 29. Pyramid: base area 3850 ft2, height 125 ft 30. Sphere: diameter 2.21 mm 31. Cone: base radius 32.4 cm, height 50.7 cm 32. Frustum of a cone: base radius 2.336 ft, top radius 2.016 ft, height
4.890 ft
In Exercises 33–36, find the surface area of the indicated solid geometric figure.
33. Total area of cube of edge 0.520 m 34. Total area of cylinder: base diameter 12.0 ft, height 58.0 ft 35. Lateral area of cone: base radius 2.56 in., height 12.3 in. 36. Total area of sphere: d = 12,760 km
C In Exercises 37–40; use Fig. 2.133. Line CT is tangent to the circle with center at O. Find the indicated angles.
37. ∠BTA
38. ∠TAB
T
39. ∠BTC
40. ∠ABT
B 50° A
O
Fig. 2.133
In Exercises 41–44, use Fig. 2.134. Given that AB = 4, BC = 4, CD = 6, and ∠ADC = 53°, find the indicated angle and lengths.
41. ∠ABE
C
42. AD
B
43. BE
44. AE
A
Fig. 2.134
E
D
In Exercises 45–48, find the formulas for the indicated perimeters and areas.
45. Perimeter of Fig. 2.135 (a right triangle and semicircle attached)
46. Perimeter of Fig. 2.136 (a square with a quarter circle at each end)
47. Area of Fig. 2.135
48. Area of Fig. 2.136
52. If the dimensions of a solid geometric figure are each multiplied by n, by how much is the volume multiplied? Explain, using a cube to illustrate.
53. What is an equation relating chord segments a, b, c, and d shown in Fig. 2.137. The dashed chords are an aid in the solution.
ab
c d
Fig. 2.137
54. From a common point, two line segments are tangent to the same circle. If the angle between the line segments is 36°, what is the angle between the two radii of the circle drawn from the points of tangency?
In Exercises 55–84, solve the given problems.
55. A tooth on a saw is in the shape of an isosceles triangle. If the angle at the point is 32°, find the two base angles.
56. A lead sphere 1.50 in. in diameter is flattened into a circular sheet 14.0 in. in diameter. How thick is the sheet?
57. A ramp for the disabled is designed so that it rises 0.48 m over a horizontal distance of 7.8 m. How long is the ramp?
58. An airplane is 2100 ft directly above one end of a 9500-ft runway. How far is the plane from the glide-slope indicator on the ground at the other end of the runway?
59. A machine part is in the shape of a square with equilateral triangles attached to two sides (see Fig. 2.138). Find the perimeter of the machine part.
p = 18.0 m (for square)
2.4 cm
Fig. 2.138
Fig. 2.139
60. A patio is designed with semicircular areas attached to a square, as shown in Fig. 2.139. Find the area of the patio.
61. A cell phone transmitting tower is supported by guy wires. The tower and three parallel guy wires are shown in Fig. 2.140. Find the distance AB along the tower.
2a b
Fig. 2.135
s Fig. 2.136
In Exercises 49–54, answer the given questions.
49. Is a square also a rectangle, a parallelogram, and a rhombus? 50. If the measures of two angles of one triangle equal the measures
of two angles of a second triangle, are the two triangles similar? 51. If the dimensions of a plane geometric figure are each multiplied
by n, by how much is the area multiplied? Explain, using a circle to illustrate.
14 m B A
B A
13 m 18 m Fig. 2.140
Main Street Fig. 2.141
62. Find the areas of lots A and B in Fig. 2.141. A has a frontage on Main St. of 140 ft, and B has a frontage on Main St. of 84 ft. The boundary between lots is 120 ft.
First Street
Review Exercises
83
63. To find the height of a flagpole, a person places a mirror at M, as shown in Fig. 2.142. The person’s eyes at E are 160 cm above the ground at A. From physics, it is known that ∠AME = ∠BMF. If AM = 120 cm and MB = 4.5 m, find the height BF of the flagpole.
F
E Mirror
1
2
A
M
B
Fig. 2.142
64. A computer screen displays a circle inscribed in a square and a square inscribed in the circle. Find the ratio of (a) the area of the inner square to the area of the outer square, (b) the perimeter of the inner square to the perimeter of the outer square.
65. A typical scale for an aerial photograph is 1>18450. In an 8.00-by 10.0-in. photograph with this scale, what is the longest distance (in mi) between two locations in the photograph?
66. For a hydraulic press, the mechanical advantage is the ratio of the large piston area to the small piston area. Find the mechanical advantage if the pistons have diameters of 3.10 cm and 2.25 cm.
67. The diameter of the Earth is 7920 mi, and a satellite is in orbit at an altitude of 210 mi. How far does the satellite travel in one rotation about the Earth?
68. The roof of the Louisiana Superdome in New Orleans is supported by a circular steel tension ring 651 m in circumference. Find the area covered by the roof.
69. A rectangular piece of wallboard with two holes cut out for heating ducts is shown in Fig. 2.143. What is the area of the remaining piece? 8.0 f t
4.0 f t
1.0 f t 1.0 f t
72. To build a highway, it is necessary to cut through a hill. A surveyor measured the cross-sectional areas at 250-ft intervals through the cut as shown in the following table. Using the trapezoidal rule, determine the volume of soil to be removed.
Dist. (ft)
0 250 500 750 1000 1250 1500 1750
Area 1ft22 560 1780 4650 6730 5600 6280 2260 230
73. The Hubble space telescope is within a cylinder 4.3 m in diameter and 13 m long. What is the volume within this cylinder?
74. A horizontal cross section of a concrete bridge pier is a regular hexagon (six sides, all equal in length, and all internal angles are equal), each side of which is 2.50 m long. If the height of the pier is 6.75 m, what is the volume of concrete in the pier?
75. A railroad track 1000.00 ft long expands 0.20 ft (2.4 in.) during the afternoon (due to an increase in temperature of about 30°F). Assuming that the track cannot move at either end and that the increase in length causes a bend straight up in the middle of the track, how high is the top of the bend?
76. On level ground a straight guy wire is attached to a vertical antenna. The guy wire is anchored in the ground 15.6 ft from the base of the antenna, and is 4.0 ft longer than the distance up the pole where it is attached. How long is the guy wire?
77. On a straight east-west road, a man walks 1500 m to the east and a woman walks 600 m to the west until they meet. They turn south and walk to a point that is 1700 m (on a straight line) from his starting point. How far is she (on a straight line) from her starting point?
78. A basketball court is 44 ft longer than it is wide. If the perimeter is 288 ft, what are the length and width?
79. A hot-water tank is in the shape of a right circular cylinder surmounted by a hemisphere as shown in Fig. 2.145. How many gallons does the tank hold? (1.00 ft3 contains 7.48 gal.)
Fig. 2.143
70. The diameter of the sun is 1.38 * 106 km, the diameter of the Earth is 1.27 * 104 km, and the distance from the Earth to the sun (center to center) is 1.50 * 108 km. What is the distance from the center of the Earth to the end of the shadow due to the rays from the sun?
71. Using aerial photography, the width of an oil spill is measured at 250-m intervals, as shown in Fig. 2.144. Using Simpson’s rule, find the area of the oil spill.
4.75 f t 2.50 f t
220 m 530 m
480 m 320 m
510 m 350 m 730 m 560 m 240 m
190 m
260 m Fig. 2.144
250 m
Fig. 2.145
80. A tent is in the shape of a regular pyramid surmounted on a cube. If the edge of the cube is 2.50 m and the total height of the tent is 3.25 m, find the area of the material used in making the tent (not including any floor area).
84
CHAPTER 2 Geometry
81. The diagonal of a TV screen is 152 cm. If the ratio of the width of the screen to the height of the screen is 16 to 9, what are the width and height?
82. An avid math student wrote to a friend, “My sailboat has a right triangular sail with edges (in ft) of 3k - 1, 4k + 3, and 5k + 2. Can you tell me the area of the sail if k 7 1?”
83. The friend in Exercise 82 replied, “Yes, and I have two cups that hold exactly the same amount of water, one of which is cylindrical and the other is hemispherical. If I tell you the height of the cylinder, can you tell me the radius of each, if the radii are equal?”
84. A satellite is 1590 km directly above the center of the eye of a circular (approximately) hurricane that has formed in the Atlantic Ocean. The distance from the satellite to the edge of the hurricane is 1620 km. What area does the hurricane cover? Neglect the curvature of the Earth and any possible depth of the hurricane.
85. The Pentagon, headquarters of the U.S. Department of Defense, is the world’s largest office building. It is a regular pentagon (five sides, all equal in length, and all interior angles are equal) 921 ft on a side, with a diagonal of length 1490 ft. Using these data, draw a sketch and write one or two paragraphs to explain how to find the area covered within the outside perimeter of the Pentagon. (What is the area?)
CHAPTER 2 PRACTICE TEST
As a study aid, we have included complete solutions for each Practice Test problem at the back of this book.
1. In Fig. 2.146, determine ∠1. 2. In Fig. 2.146, determine ∠2.
2
C
D
A Fig. 2.146 52°
1 B
AB CD
3. A tree is 8.0 ft high and casts a shadow 10.0 ft long. At the same time, a telephone pole casts a shadow 25.0 ft long. How tall is the pole?
4. Find the area of a triangular wall pennant of sides 24.6 cm, 36.5 cm, and 40.7 cm.
5. What is the diagonal distance between corners of a rectangular field 125 ft wide and 170 ft long?
6. An office building hallway floor is designed in the trapezoidal shape shown in Fig. 2.147. What is the area of the hallway?
3.12 m
4.70 m 2.76 m
4.70 m
9.96 m Fig. 2.147
7. What is (a) the mass (in kg) of a cubical block of ice, the edge of which is 0.40 m (the density of ice is 0.92 * 103 kg/m3), and (b) the surface area of the block?
8. Find the surface area of a tennis ball whose circumference is 21.0 cm.
9. Find the volume of a right circular cone of radius 2.08 m and height 1.78 m.
10. In Fig. 2.148, find ∠1.
11. In Fig. 2.148, find ∠2.
64° C
1
2
A O
B
Fig. 2.148
2.25 cm Fig. 2.149
12. In Fig. 2.149, find the perimeter of the figure shown. It is a square with a semicircle removed.
13. In Fig. 2.149, find the area of the figure shown.
14. The width of a marshy area is measured at 50-ft intervals, with the results shown in the following table. Using the trapezoidal rule, find the area of the marsh. (All data accurate to two or more significant digits.)
Distance (ft) Width (ft)
0 50 100 150 200 250 300 0 90 145 260 205 110 20