Soo t tan finite mathematics for the managerial, life, and social sciences brooks cole (2008)

Preface xi 1.1 The Cartesian Coordinate System 2 1.2 Straight Lines 9 Using Technology: Graphing a Straight Line 23 1.3 Linear Functions and Mathematical Models 26 Using Technology: Eval

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Upon entering the University of Kansas as an undergraduate, Chris Shannon knew she enjoyed

mathematics, but she was also interested in a variety of social and political issues One of her

mathematics professors recognized this and suggested that she might be interested in taking

some economics courses while she was studying mathematics She learned that economics

enabled her to combine the rigor and abstraction of mathematics with the exploration of complex

and important social issues involving human behavior She decided to add a major in economics

to her math major After graduating with B.S degrees in economics and in mathematics, Shannon

went on to graduate school at Stanford University, where she received an M.S in mathematics

and a Ph.D in economics.

Her current position as professor in both the mathematics and economics departments at the

University of California, Berkeley, represents an ideal blend of the two fields, and allows her to

pursue work ranging from developing new tools for analyzing optimization problems to designing

new models for understanding complex financial markets The photos on the front cover of this

text represent one of her current projects, which explores new models of decision-making under

uncertainty and the effects of uncertainty on different markets.*

Look for other featured applied researchers in forthcoming titles in the Tan applied mathematics series:

About the Cover

PETER BLAIR HENRY

International Economist

Stanford University

MARK VAN DER LAAN

Biostatistician University of California, Berkeley

JONATHAN D FARLEY

Applied Mathematician California Institute

of Technology

NAVIN KHANEJA

Applied Scientist Harvard University

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r is the interest rate per year, m is the number of conversion periods per year, and t is the

number of years

where reffis the effective rate of interest, r is the nominal interest rate per year, and m is

the number of conversion periods per year

Equation of the Least-Squares Line

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FINITE MATHEMATICS FOR THE MANAGERIAL, LIFE,

AND SOCIAL SCIENCES EDITION

9

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FINITE MATHEMATICS FOR THE MANAGERIAL, LIFE,

AND SOCIAL SCIENCES

S T TAN

STONEHILL COLLEGE

EDITION9

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Finite Mathematics for the Managerial, Life, and Social Sciences, Ninth Edition

S T Tan Senior Acquisitions Editor: Carolyn Crockett Development Editor: Danielle Derbenti Assistant Editor: Catie Ronquillo Editorial Assistant: Rebecca Dashiell Technology Project Manager: Ed Costin Marketing Manager: Mandy Jellerichs Marketing Assistant: Ashley Pickering Marketing Communications Manager:

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Cover Designer: Irene Morris Cover Images: Chris Shannon © Cengage Learning; Numbers, George

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1 2 3 4 5 6 7 12 11 10 09 08

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TO PAT, BILL, AND MICHAEL

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Preface xi

1.1 The Cartesian Coordinate System 2

1.2 Straight Lines 9

Using Technology: Graphing a Straight Line 23

1.3 Linear Functions and Mathematical Models 26

Using Technology: Evaluating a Function 37

1.4 Intersection of Straight Lines 40

PORTFOLIO: Esteban Silva 42

Using Technology: Finding the Point(s) of Intersection of Two Graphs 49

1.5 The Method of Least Squares 51

Using Technology: Finding an Equation of a Least-Squares Line 60

Chapter 1 Summary of Principal Formulas and Terms 63 Chapter 1 Concept Review Questions 64

Chapter 1 Review Exercises 65 Chapter 1 Before Moving On 66

2.1 Systems of Linear Equations: An Introduction 68

2.2 Systems of Linear Equations: Unique Solutions 75

Using Technology: Systems of Linear Equations: Unique Solutions 89

2.3 Systems of Linear Equations: Underdetermined and Overdetermined Systems 91

Using Technology: Systems of Linear Equations: Underdetermined and

Overdetermined Systems 100

2.4 Matrices 100

Using Technology: Matrix Operations 110

2.5 Multiplication of Matrices 113

Using Technology: Matrix Multiplication 125

2.6 The Inverse of a Square Matrix 127

Using Technology: Finding the Inverse of a Square Matrix 139

2.7 Leontief Input–Output Model 141

Using Technology: The Leontief Input–Output Model 149

3.1 Graphing Systems of Linear Inequalities in Two Variables 156

3.2 Linear Programming Problems 164

3.3 Graphical Solution of Linear Programming Problems 172

Straight Lines and Linear Functions 1

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3.4 Sensitivity Analysis 185

PORTFOLIO: Morgan Wilson 192

Chapter 3 Summary of Principal Terms 198 Chapter 3 Concept Review Questions 198 Chapter 3 Review Exercises 198

Chapter 3 Before Moving On 200

4.1 The Simplex Method: Standard Maximization Problems 202

Using Technology: The Simplex Method: Solving Maximization Problems 222

4.2 The Simplex Method: Standard Minimization Problems 226

Using Technology: The Simplex Method: Solving Minimization Problems 237

4.3 The Simplex Method: Nonstandard Problems 242

Chapter 4 Summary of Principal Terms 254 Chapter 4 Concept Review Questions 254 Chapter 4 Review Exercises 255 Chapter 4 Before Moving On 256

5.1 Compound Interest 258

Using Technology: Finding the Accumulated Amount of an Investment, the

Effective Rate of Interest, and the Present Value of an Investment 273

5.2 Annuities 276

Using Technology: Finding the Amount of an Annuity 284

5.3 Amortization and Sinking Funds 287

Using Technology: Amortizing a Loan 297

5.4 Arithmetic and Geometric Progressions 300

6.1 Sets and Set Operations 314

6.2 The Number of Elements in a Finite Set 323

6.3 The Multiplication Principle 329

PORTFOLIO: Stephanie Molina 331

6.4 Permutations and Combinations 335

Using Technology: Evaluating n!, P(n, r), and C(n, r) 348

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7.1 Experiments, Sample Spaces, and Events 354

7.2 Definition of Probability 362

7.3 Rules of Probability 371

PORTFOLIO: Todd Good 374

7.4 Use of Counting Techniques in Probability 381

7.5 Conditional Probability and Independent Events 388

8.1 Distributions of Random Variables 418

Using Technology: Graphing a Histogram 425

8.2 Expected Value 427

PORTFOLIO: Ann-Marie Martz 434

8.3 Variance and Standard Deviation 440

Using Technology: Finding the Mean and Standard Deviation 451

8.4 The Binomial Distribution 452

8.5 The Normal Distribution 462

8.6 Applications of the Normal Distribution 471

9.1 Markov Chains 484

Using Technology: Finding Distribution Vectors 493

9.2 Regular Markov Chains 494

Using Technology: Finding the Long-Term Distribution Vector 503

9.3 Absorbing Markov Chains 505

9.4 Game Theory and Strictly Determined Games 512

9.5 Games with Mixed Strategies 521

A.1 Propositions and Connectives 538

A.2 Truth Tables 542

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A.3 The Conditional and the Biconditional Connectives 544

A.4 Laws of Logic 549

A.5 Arguments 553

A.6 Applications of Logic to Switching Networks 558

Table 1: Binomial Probabilities 574

Table 2: The Standard Normal Distribution 577

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as statistics and operations research, and (3) to make the text a useful tool for instructors.The only prerequisite for understanding this text is 1 to 2 years, or the equivalent, of highschool algebra.

This text offers more than enough material for a one-semester or two-quarter course.The following chart on chapter dependency is provided to help the instructor design acourse that is most suitable for the intended audience

1

Straight Lines and

Linear Functions

6

Sets and Counting

9

Markov Chains and the Theory

A Geometric Approach

8

Probability Distributions and Statistics

4

Linear Programming:

An Algebraic Approach

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xii PREFACE

Level of Presentation

My approach is intuitive, and I state the results informally However, I have taken special care

to ensure that this approach does not compromise the mathematical content and accuracy

Problem-Solving Approach

A problem-solving approach is stressed throughout the book Numerous examples andapplications illustrate each new concept and result Special emphasis is placed on helpingstudents formulate, solve, and interpret the results of the problems involving applications.Graphs and illustrations are used extensively to help students visualize the concepts andideas being presented

Motivating Real-World

Applications

More than 90 new applications have

been added Among these applications

are U.S health-care expenditures,

satellite TV subscribers, worldwide

consulting spending, investment

portfolios, adjustable-rate mortgages,

green companies, security breaches,

distracted drivers, obesity in children,

and water supply

NEW TO THIS EDITION

Modeling with Data

Students can actually see how some

of the functions found in the examples

and exercises are constructed (See

Applied Example 1, U.S Health-Care

Expenditures, page 29, and the

corre-sponding example from which the model

is derived in Applied Example 3, page

55.) Modeling with Data exercises are

now found in Using Technology,

Section 1.5

THE APPROACH

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Making Connections with Technology

A new example—Market forCholesterol-Reducing Drugs—

has been added to UsingTechnology 1.3 Also,Exploring with Technologyexamples illustrating the use ofthe graphing calculator to solveinequalities, to generate randomnumbers, and to find the areaunder the standard normal curvehave been added

Using Logarithms to Solve Problems in Finance—New Optional Examples and Exercises

This new subsection has been added to Chapter 5,Mathematics of Finance

Examples and exercises inwhich the rate of interest issolved for, or the time needed tomeet an investment goal isfound, are now covered here

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xiv PREFACE

Action-Oriented

Study Tabs

Convenient color-coded study

tabs, similar to Post-it®flags,

make it easy for students to tab

pages that they want to return to

later, whether it be for

additional review, exam

preparation, online exploration,

or identifying a topic to be

discussed with the instructor

Specific Content Changes

■ A new mathematical model, U.S health-care expenditures, is discussed in Section 1.3

In Section 1.5, the linear function used in the model is derived using the least-squaresmethod

■ The discussion of mortgages has been enhanced with a new example on adjustable-ratemortgages and the addition of many new applied exercises

■ More rote and applied exercises have been added to the chapter reviews

■ Appendix A, on Logic, has been revised

■ A Review of Logarithms is now found in Appendix C This material supplements theoptional subsection on Using Logarithms to Solve Problems in Finance in Chapter 5

■ A How-To Technology Index has been added for easy reference

■ The complete solutions to the exercises in Appendix A, Logic, have been added to the

Instructor’s Solutions Manual, and the odd-numbered solutions for these exercises have been added to the Student Solutions Manual.

■ New Using Technology Excel sections for Microsoft Office 2007 will now be available

on the Web

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In addition to the new features, we have retained many of the following hallmarks thathave made this series so usable and well-received in past editions:

■ Section exercises to help students understand and apply concepts

■ Optional technology sections to explore mathematical ideas and solve problems

■ End-of-chapter review sections to assess understanding and problem-solving skills

■ Features to motivate further exploration

Self-Check Exercises

Offering students immediatefeedback on key concepts, theseexercises begin each end ofsection exercise set Fullyworked-out solutions can befound at the end of eachexercise section

Concept Questions

Designed to test students’

understanding of the basicconcepts discussed in thesection, these questionsencourage students to explainlearned concepts in their ownwords

Exercises

Each exercise section contains

an ample set of problems of aroutine computational naturefollowed by an extensive set ofapplication-oriented problems

TRUSTED FEATURES

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xvi PREFACE

Using Technology

These optional subsections

appear after the section

exercises They can be used in

the classroom if desired or as

material for self-study by the

student Here, the graphing

calculator and Microsoft Excel

2003 are used as a tool to solve

problems (Instructions for

Microsoft Excel 2007 are given

at the Companion Website.)

These sections are written in the

traditional example–exercise

format, with answers given at

the back of the book

Illustrations showing graphing

calculator screens and

spreadsheets are extensively

used In keeping with the theme

of motivation through real-life

examples, many sourced

applications are again included

Students can construct their own

models using real-life data in

Using Technology Section 1.5

Exploring with Technology

Designed to explore mathematical concepts and

to shed further light on examples in the text,these optional discussions appear throughout themain body of the text and serve to enhance thestudent’s understanding of the concepts andtheory presented Often the solution of anexample in the text is augmented with a

graphical or numerical solution Completesolutions to any questions posed are given in

the Instructor’s Solutions Manual.

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Summary of Principal Formulas and Terms

Each review section begins withthe Summary highlighting impor-tant equations and terms with pagenumbers given for quick review

Concept Review Questions

These questions give students achance to check their knowl-edge of the basic definitions andconcepts given in each chapter

Review Exercises

Offering a solid review of the chaptermaterial, the Review Exercises containroutine computational exercises followed

by applied problems

Before Moving On

Found at the end of each chapterreview, these exercises give students achance to see if they have masteredthe basic computational skillsdeveloped in each chapter If theysolve a problem incorrectly, they can

go to the Companion Website and tryagain In fact, they can keep on tryinguntil they get it right If students needstep-by-step help, they can use the

CengageNOW Tutorials that are

keyed to the text and work out similarproblems at their own pace

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xviii PREFACE

Explore & Discuss

These optional questions can be

discussed in class or assigned as

homework These questions generally

require more thought and effort than the

usual exercises They may also be used

to add a writing component to the class

or as team projects Complete solutions

to these exercises are given in the

Instructor’s Solutions Manual

Portfolios

The real-life experiences of a variety

of professionals who use

mathematics in the workplace are

related in these interviews Among

those interviewed are a senior

account manager at PepsiCo and an

associate on Wall Street who uses

statistics and calculus in writing

options

Example Videos

Available through the Online

Resource Center and Enhanced

WebAssign, these video

examples offer hours of

instruction from award-winning

teacher Deborah Upton of

Stonehill College Watch as she

walks students through key

examples from the text, step by

step—giving them a foundation in the skills that they need to know Eachexample available online is identified by the video icon located in the margin

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INSTRUCTOR’S SOLUTIONS MANUAL(ISBN 0-495-38897-1) by Soo T TanThe complete solutions manual provides worked out solutions to all problems in the text,

as well as “Exploring with Technology” and “Explore & Discuss” questions

POWERLECTURE(ISBN 0-495-38899-8)

This comprehensive CD-ROM includes the Instructor’s Solutions Manual, PowerPoint

Slides, and ExamView® Computerized Testing featuring algorithmically generated tions to create, deliver, and customize tests

ques-ENHANCED WEBASSIGN

Instant feedback and ease of use are just two reasons why WebAssign is the most widelyused homework system in higher education WebAssign allows you to assign, collect,grade, and record homework assignments via the Web Now this proven homework sys-tem has been enhanced to include links to textbook sections, video examples, and prob-lem-specific tutorials Enhanced WebAssign is more than a homework system—it is acomplete learning system for math students

I wish to express my personal appreciation to each of the following reviewers of the EighthEdition, whose many suggestions have helped make a much improved book

Miami Dade College

Stephanie Anne Salomone

as problem-solving strategies, additional algebra steps, and review for selected problems

ONLINE RESOURCE CENTER(ISBN 0-495-56386-4)Sign in, save time, and get the grade you want! One code will give you access to great tools

for Finite Mathematics for the Managerial, Life, and Social Sciences, Ninth Edition It

includes Personal Tutor (online tutoring with an expert that offers help right now),

CengageNOW (an online diagnostic, homework, and tutorial system), and access to new

Solution Videos on the password-protected Premium Website

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I also wish to thank Jerrold Grossman and Tao Guo for their many helpful suggestionsfor improving the text I am also grateful to Jerrold for the superb job he did as the accu-racy checker for this text A special thanks goes to Jill Britton for contributing some of thenew linear programming problems for this edition I also thank the editorial, production,and marketing staffs of Brooks/Cole: Carolyn Crockett, Danielle Derbenti, Catie Ronquillo,Cheryll Linthicum, Mandy Jellerichs, Sam Subity, Jennifer Liang, and Rebecca Dashiell forall of their help and support during the development and production of this edition I alsothank Martha Emry and Betty Duncan who both did an excellent job of ensuring the accu-racy and readability of this edition Simply stated, the team I have been working with is out-standing, and I truly appreciate all their hard work and efforts Finally, a special thanks tothe mathematicians—Chris Shannon and Mark van der Lann at Berkeley, Peter Blair Henry

at Stanford, Jonathan D Farley at Cal Tech, and Navin Khaneja at Harvard—for takingtime off from their busy schedules to describe how mathematics is used in their research.Their pictures and applications of their research appear on the covers of my applied math-ematics series

Jacksonville State University

Sandra Pryor Clarkson

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A BOUT THE A UTHOR

SOO T TAN received his S.B degree from Massachusetts Institute

of Technology, his M.S degree from the University ofWisconsin–Madison, and his Ph.D from the University of California

at Los Angeles He has published numerous papers in Optimal Control Theory, Numerical Analysis, and Mathematics of Finance He

is currently a Professor of Mathematics at Stonehill College

By the time I started writing the first of what turned out to be a series of textbooks in mathematics for students in the managerial, life, and social sciences, I had quite a few years of experience teaching mathematics to non-mathematics majors One of the most important lessons I learned from my early experience teaching these courses is that many

of the students come into these courses with some degree of apprehension This ness led to the intuitive approach I have adopted in all of my texts As you will see, I try to introduce each abstract mathematical concept through an example drawn from a com- mon, real-life experience Once the idea has been conveyed, I then proceed to make it precise, thereby assuring that no mathematical rigor is lost in this intuitive treatment of the subject Another lesson I learned from my students is that they have a much greater appreciation of the material if the applications are drawn from their fields of interest and from situations that occur in the real world This is one reason you will see so many exercises in my texts that are modeled on data gathered from newspapers, magazines, jour- nals, and other media Whether it be the market for cholesterol-reducing drugs, financing

aware-a home, bidding for caware-able rights, broaware-adbaware-and Internet households, or Staware-arbucks’ aware-annuaware-al sales, I weave topics of current interest into my examples and exercises to keep the book relevant to all of my readers.

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Because the over-65 population will

be growing more rapidly in the next few decades, U.S health-care expenditures are expected to be boosted significantly What will be the rate of increase of these expenditures over the next few years? How much will health-care expenditures be in 2011? In Example 1, page 29, we use a mathematical model based on figures from the Centers for Medicare and Medicaid to answer these questions.

system, a system that allows us to represent points in theplane in terms of ordered pairs of real numbers This in turnenables us to compute the distance between two points

algebraically We also study straight lines Linear functions, whose

graphs are straight lines, can be used to describe manyrelationships between two quantities These relationships can befound in fields of study as diverse as business, economics, thesocial sciences, physics, and medicine In addition, we see howsome practical problems can be solved by finding the point(s) ofintersection of two straight lines Finally, we learn how to find analgebraic representation of the straight line that “best” fits a set

of data points that are scattered about a straight line

STRAIGHT LINES AND

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2 1 STRAIGHT LINES AND LINEAR FUNCTIONS

The Cartesian Coordinate System

The real number system is made up of the set of real numbers together with the usualoperations of addition, subtraction, multiplication, and division We assume that youare familiar with the rules governing these algebraic operations (see Appendix B).Real numbers may be represented geometrically by points on a line This line is

called the real number, or coordinate, line We can construct the real number line as

follows: Arbitrarily select a point on a straight line to represent the number 0 This

point is called the origin If the line is horizontal, then choose a point at a convenient

distance to the right of the origin to represent the number 1 This determines the scale

for the number line Each positive real number x lies x units to the right of 0, and each negative real number x lies x units to the left of 0.

In this manner, a one-to-one correspondence is set up between the set of real bers and the set of points on the number line, with all the positive numbers lying to theright of the origin and all the negative numbers lying to the left of the origin (Figure 1)

num-In a similar manner, we can represent points in a plane (a two-dimensional space)

by using the Cartesian coordinate system , which we construct as follows: Take two

perpendicular lines, one of which is normally chosen to be horizontal These lines

intersect at a point O, called the origin (Figure 2) The horizontal line is called the

x -axis, and the vertical line is called the y-axis A number scale is set up along the

x-axis, with the positive numbers lying to the right of the origin and the negative

num-bers lying to the left of it Similarly, a number scale is set up along the y-axis, with the

positive numbers lying above the origin and the negative numbers lying below it

Note The number scales on the two axes need not be the same Indeed, in many

applications different quantities are represented by x and y For example, x may resent the number of cell phones sold and y the total revenue resulting from the sales.

rep-In such cases it is often desirable to choose different number scales to represent thedifferent quantities Note, however, that the zeros of both number scales coincide atthe origin of the two-dimensional coordinate system

We can represent a point in the plane uniquely in this coordinate system by an

ordered pairof numbers—that is, a pair (x, y), where x is the first number and y the second To see this, let P be any point in the plane (Figure 3) Draw perpendiculars from P to the x-axis and y-axis, respectively Then the number x is precisely the number that corresponds to the point on the x-axis at which the perpendicular through P hits the x-axis Similarly, y is the number that corresponds to the point on the y-axis at which the perpendicular through P crosses the y-axis.

Conversely, given an ordered pair (x, y) with x as the first number and y the ond, a point P in the plane is uniquely determined as follows: Locate the point on the

sec-x -axis represented by the number x and draw a line through that point perpendicular

to the x-axis Next, locate the point on the y-axis represented by the number y and draw

a line through that point perpendicular to the y-axis The point of intersection of these two lines is the point P (Figure 3).

An ordered pair in the coordinate plane

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In the ordered pair (x, y), x is called the abscissa, or x-coordinate, y is called the ordinate, or y-coordinate, and x and y together are referred to as the coordinatesof

the point P The point P with x-coordinate equal to a and y-coordinate equal to b is often written P(a, b).

The points A(2, 3), B( 2, 3), C(2, 3), D(2, 3), E(3, 2), F(4, 0), and

G(0, 5) are plotted in Figure 4

Note In general, (x, y) (y, x) This is illustrated by the points A and E in Figure 4.

The axes divide the plane into four quadrants Quadrant I consists of the points P with coordinates x and y, denoted by P(x, y), satisfying x 0 and y 0; Quadrant II, the points P(x, y) where x 0 and y 0; Quadrant III, the points P(x, y) where x 0 and

y 0; and Quadrant IV, the points P(x, y) where x 0 and y 0 (Figure 5).

The Distance Formula

One immediate benefit that arises from using the Cartesian coordinate system is thatthe distance between any two points in the plane may be expressed solely in terms of

the coordinates of the points Suppose, for example, (x1, y1) and (x2, y2) are any twopoints in the plane (Figure 6) Then we have the following:

For a proof of this result, see Exercise 46, page 8

x O

Quadrant I (+, +)

y

Quadrant II (–, +)

Quadrant III (–, –)

Quadrant IV (+, –)

x

y

4 2

–3 –1

D(2, – 3) C(–2, –3)

– 2 – 4 – 6

F(4, 0) E(3, 2)

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In what follows, we give several applications of the distance formula.

EXAMPLE 1 Find the distance between the points (4, 3) and (2, 6)

Solution Let P1(4, 3) and P2(2, 6) be points in the plane Then we have

x1 4 and y1 3

x2 2 y2 6Using Formula (1), we have

APPLIED EXAMPLE 2 The Cost of Laying Cable In Figure 7, S

rep-resents the position of a power relay station located on a straight coastal

highway, and M shows the location of a marine biology experimental station on a nearby island A cable is to be laid connecting the relay station at S with the experimental station at M via the point Q that lies on the x-axis between O and S If the

cost of running the cable on land is $3 per running foot and the cost of running thecable underwater is $5 per running foot, find the total cost for laying the cable

Solution The length of cable required on land is given by the distance from S

to Q This distance is (10,000 2000), or 8000 feet Next, we see that the length

of cable required underwater is given by the distance from Q to M This distance is

or approximately 3605.55 feet Therefore, the total cost for laying the cable is

3(8000) 5(3605.55) ⬇ 42,027.75

or approximately $42,027.75

EXAMPLE 3 Let P(x, y) denote a point lying on a circle with radius r and center C(h, k) (Figure 8) Find a relationship between x and y.

Solution By the definition of a circle, the distance between C(h, k) and P(x, y)

is r Using Formula (1), we have

The cable will connect the relay station S

to the experimental station M.

Explore & Discuss

Refer to Example 1 Suppose

we label the point (2, 6) as

P1 and the point ( 4, 3) as

P2 (1) Show that the

dis-tance d between the two

points is the same as that

obtained earlier (2) Prove

that, in general, the distance

d in Formula (1) is

indepen-dent of the way we label the

two points.

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which, upon squaring both sides, gives the equation

(x h)2 (y k)2 r2

which must be satisfied by the variables x and y.

A summary of the result obtained in Example 3 follows

EXAMPLE 4 Find an equation of the circle with (a) radius 2 and center (1, 3)and (b) radius 3 and center located at the origin

– 1

(–1, 3) 2

x

y

3 1

Explore & Discuss

1 Use the distance formula to help you describe the set of points in the xy-plane satisfying

each of the following inequalities, where r 0.

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1 a Plot the points A(4, 2), B(2, 3), and C(3, 1).

b Find the distance between the points A and B, between

B and C, and between A and C.

c Use the Pythagorean theorem to show that the triangle

with vertices A, B, and C is a right triangle.

2 The accompanying figure shows the location of cities A, B,

and C Suppose a pilot wishes to fly from city A to city C

but must make a mandatory stopover in city B If the

single-1 What can you say about the signs of a and b if the point

P(a, b) lies in: (a) The second quadrant? (b) The third

quadrant? (c) The fourth quadrant?

2 a What is the distance between P1(x1, y1) and P2(x2, y2)?

b When you use the distance formula, does it matter

which point is labeled P1and which point is labeled

B(200, 50) A(0, 0)

engine light plane has a range of 650 mi, can the pilot make

the trip without refueling in city B?

In Exercises 1–6, refer to the accompanying figure and

determine the coordinates of the point and the quadrant

F

E

In Exercises 7–12, refer to the accompanying figure.

7 Which point has coordinates (4, 2)?

8 What are the coordinates of point B?

9 Which points have negative y-coordinates?

10 Which point has a negative x-coordinate and a negative

y-coordinate?

11 Which point has an x-coordinate that is equal to zero?

12 Which point has a y-coordinate that is equal to zero?

x

y

4 2

F E C

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In Exercises 13–20, sketch a set of coordinate axes and then plot the point.

25 Find the coordinates of the points that are 10 units away

from the origin and have a y-coordinate equal to 6

26 Find the coordinates of the points that are 5 units away

from the origin and have an x-coordinate equal to 3.

27 Show that the points (3, 4), (3, 7), (6, 1), and (0, 2)form the vertices of a square

28 Show that the triangle with vertices (5, 2), (2, 5), and(5,2) is a right triangle

In Exercises 29–34, find an equation of the circle that satisfies the given conditions.

29 Radius 5 and center (2, 3)

30 Radius 3 and center (2, 4)

31 Radius 5 and center at the origin

32 Center at the origin and passes through (2, 3)

33 Center (2, 3) and passes through (5, 2)

34 Center (a, a) and radius 2a

35 D ISTANCE T RAVELED A grand tour of four cities begins at city

A and makes successive stops at cities B, C, and D before

returning to city A If the cities are located as shown in the

accompanying figure, find the total distance covered on the tour.

36 DELIVERY CHARGES A furniture store offers free setup anddelivery services to all points within a 25-mi radius of itswarehouse distribution center If you live 20 mi east and

14 mi south of the warehouse, will you incur a deliverycharge? Justify your answer

37 OPTIMIZING TRAVEL TIME Towns A, B, C, and D are

located as shown in the accompanying figure Two

high-ways link town A to town D Route 1 runs from town A to town D via town B, and Route 2 runs from town A to town

D via town C If a salesman wishes to drive from town A

to town D and traffic conditions are such that he could

expect to average the same speed on either route, whichhighway should he take in order to arrive in the shortesttime?

38 M INIMIZING S HIPPING C OSTS Refer to the figure for cise 37 Suppose a fleet of 100 automobiles are to be

Exer-shipped from an assembly plant in town A to town D.

They may be shipped either by freight train along Route 1

at a cost of 44¢/mile/automobile or by truck along Route

2 at a cost of 42¢/mile/automobile Which means of portation minimizes the shipping cost? What is the netsavings?

trans-39 CONSUMER DECISIONS Will Barclay wishes to determinewhich HDTV antenna he should purchase for his home The

TV store has supplied him with the following information:

Will wishes to receive Channel 17 (VHF), which is located

25 mi east and 35 mi north of his home, and Channel 38(UHF), which is located 20 mi south and 32 mi west of hishome Which model will allow him to receive both chan-nels at the least cost? (Assume that the terrain betweenWill’s home and both broadcasting stations is flat.)

40 COST OF LAYING CABLE In the accompanying diagram, S

rep-resents the position of a power relay station located on a

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To see how a straight line in the xy-plane may be described algebraically, we need

first to recall certain properties of straight lines

Slope of a Line

Let L denote the unique straight line that passes through the two distinct points (x1, y1) and (x2, y2) If x1 x2, then L is a vertical line, and the slope is undefined (Fig- ure 11) If x1 x2, then we define the slope of L as follows.

Observe that the slope of a straight line is a constant whenever it is defined

The number y y2 y1(y is read “delta y”) is a measure of the vertical change

in y, and x x2 x1is a measure of the horizontal change in x as shown in Figure

12 From this figure we can see that the slope m of a straight line L is a measure of the rate of change of y with respect to x Furthermore, the slope of a nonvertical straight

line is constant, and this tells us that this rate of change is constant

Figure 13a shows a straight line L1with slope 2 Observe that L1has the property

that a 1-unit increase in x results in a 2-unit increase in y To see this, let x 1 in

Slope of a Nonvertical Line

If (x1, y1) and (x2, y2) are any two distinct points on a nonvertical line L, then the slope m of L is given by

(x2, y2)

L

y2 – y1 = Δy

x2– x1= Δ x

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1 a The points are plotted in the following figure.

b The distance between A and B is

The distance between B and C is

The distance between A and C is

c We will show that

[d(A, C)]2 [d(A, B)]2 [d(B, C)]2

From part (b), we see that [d(A, B)]2 29, [d(B, C)]2

29, and [d(A, C )]2 58, and the desired result follows

2 The distance between city A and city B is

or 206 mi The distance between city B and city C is

or 483 mi Therefore, the total distance the pilot would

have to cover is 689 mi, so she must refuel in city B.

server that has an initial value of $10,000 and that is being depreciated linearly over

5 years with a scrap value of $3,000 Note that only the solid portion of the straightline is of interest here

The book value of the server at the end of year t, where t lies between 0 and 5,

can be read directly from the graph But there is one shortcoming in this approach: Theresult depends on how accurately you draw and read the graph A better and more

accurate method is based on finding an algebraic representation of the depreciation

line (We continue our discussion of the linear depreciation problem in Section 1.3.)

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