Mathematical Thinking and Quantitative Reasoning PDF
Preview Mathematical Thinking and Quantitative Reasoning
Mathematical Thinking and Quantitative Reasoning Richard N. Aufmann Palomar College, California Joanne S. Lockwood New Hampshire Community Technical College, New Hampshire Richard D. Nation Palomar College, California Daniel K. Clegg Palomar College, California HOUGHTON MIFFLIN COMPANY Boston New York Publisher: Richard Stratton Senior Sponsoring Editor: Lynn Cox Senior Marketing Manager: Katherine Greig Marketing Associate: Naveen Hariprasad Development Editor: Lisa Collette Associate Editor: Noel Kamm Editorial Assistant: Laura Ricci Associate Project Editor: Susan Miscio Editorial Assistant: Joanna Carter Art and Design Manager: Gary Crespo Cover Design Manager: Anne Katzeff Photo Editor: Jennifer Meyer Dare Composition Buyer: Chuck Dutton New Title Project Manager: James Lonergan Cover Photograph: © Royalty-Free/CORBIS Photo credits for front endpapers: Pyramid of Cheops: CORBIS; abacus: Getty Images; Great Wall of China: CORBIS; Mona Lisa: Getty Images; telescope: CORBIS; Leonhard Euler: Kean Collection/Getty Images; Declaration of Independence: CORBIS; bicycle: Getty Images; Panama Canal; Getty Images; Grace Hopper: Bettman/CORBIS; Neil Armstrong: Getty Images. Additional photo credits are found immediately after the answer section in the back of the book. Copyright © 2008 by Houghton Mifflin Company. All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system without the prior written permission of Houghton Mifflin Company unless such copying is expressly permitted by federal copyright law. Address inquiries to Houghton Mifflin, 222 Berkeley Street, Boston, MA 02116-3764. Printed in the U.S.A. Library of Congress Catalog Card Number: 2006923761 ISBNs: Instructor’s Annotated Edition: ISBN 13: 978-0-618-77738-9 ISBN 10: 0-618-77738-5 For orders, use student text ISBNs: ISBN 13: 978-0-618-77737-2 ISBN 10: 0-618-77737-7 123456789-CRK-11 10 09 08 07 APPLICATIONS Blood types 43 Color mixing 50 Counting problems 17, 18, 19, 20, 27, 28, 49, 54, 58, 59 Game strategies 1, 11, 15 Logical reasoning 2, 3, 4, 6, 8, 9, 13, 14, 20, 21, 22, 23, 24, 28, 29, 30, 31, 32, 54, 55, 56, 58 Map coloring 14, 55 Mathematics 24, 32 Physics 12, 15 Polya’s Problem-Solving Strategy 17, 18, 19, 20, 21, 22, 28 Reading and interpreting graphs 26, 30, 32, 48, 56, 59 Surveys 44, 45, 51, 52, 57, 58, 59, 60 Voting 46, 47, 49 Section 1.1 Inductive and Deductive Reasoning 2 Investigation: The Game of Sprouts by John H. Conway 10 Section 1.2 Problem-Solving Strategies 15 Investigation: Routes on a Probability Demonstrator 27 Section 1.3 Problem Solving Using Sets 33 Investigation: Voting Systems 46 Chapter Summary 53 • Review Exercises 54 • Chapter Test 58 iii CHAPTER 1 Problem Solving 1 APPLICATIONS Calculator program 87 Fallacies 105, 108 Logic gates 92, 93, 94, 95 Logic puzzles 111, 114, 116 Mathematics 84 Switching networks 69, 70, 80, 81, 82 Validity of an argument 99, 100, 101, 102, 103, 104, 107, 108, 109, 110, 111, 112, 115, 116, 119, 120 Warning circuits 70, 71, 82 Section 2.1 Logic Statements and Quantifiers 62 Investigation: Switching Networks 69 Section 2.2 Truth Tables and Applications 73 Investigation: Switching Networks—Part II 80 Section 2.3 The Conditional and Related Statements 84 Investigation: Logic Gates 92 Section 2.4 Arguments 97 Investigation: Fallacies 105 Section 2.5 Euler Diagrams 108 Investigation: Using Logic to Solve Puzzles 113 Chapter Summary 117 • Review Exercises 118 • Chapter Test 120 CHAPTER 2 Logic and Its Applications 61 C O N T E N T S iv Contents APPLICATIONS Automobiles 135, 136, 137, 148, 156, 163, 165, 171, 175, 181, 185, 190, 191, 196, 199 Business 129, 138, 149, 156, 157, 163, 170, 182, 186, 191, 196, 197, 198, 200, 201 College 136, 138, 144, 145, 148, 155, 166, 196 Compensation 138, 140, 141, 144, 152, 165, 166, 180, 190, 198 Computers 136, 138, 196 Death rates 149, 150, 177 Gardening 156, 197, 200 Government 156, 158, 164, 169, 180, 197 Health and medicine 133, 154, 156, 157, 175, 198 Housing 156, 163, 167, 196 Human behavior 128, 136, 153, 157, 158, 175, 176, 178, 200 Measurement 135, 148, 152, 156, 191 Money 129, 137, 138, 140, 141, 142, 143, 152, 154, 155, 156, 165, 166, 173, 174, 178, 191, 199 Pets 175, 201 Populations 129, 130, 153, 178, 181, 196, 197, 198 Recreation 135, 143, 164, 175, 190, 199 Science 128, 136, 156, 184, 187, 188, 189, 190, 191, 192, 193, 195, 199, 201 Space vehicles 137, 152 Sports 135, 136, 151, 152, 174, 198, 200 Work 138, 175,177, 179, 180, 190, 201 Section 3.1 First-Degree Equations and Formulas 122 Investigation: Body Mass Index 132 Section 3.2 Rate, Ratio, and Proportion 139 Investigation: Earned Run Average 151 Section 3.3 Percent 159 Investigation: Federal Income Tax 171 Section 3.4 Direct and Inverse Variation 183 Investigation: Gears and Pulleys 188 Chapter Summary 194 • Review Exercises 195 • Chapter Test 199 CHAPTER 3 Algebraic Models 121 APPLICATIONS Aquariums 216, 289, 290 Arts and crafts 210, 243, 246, 250, 253, 256, 259, 260, 262, 308 Business 210, 211, 220 Carpentry 210, 262 Construction 211, 251, 254, 260, 291, 302 Consumerism 210, 211, 220 Energy 207, 208, 218, 219, 252 Food 220, 262, 285, 290, 308, 310 Gardens 246, 259, 260, 261, 262 Geometry throughout the chapter Homes and home maintenance 215, 220, 245, 251, 252, 254, 260, 261, 262, 276, 277, 292, 301, 302, 308 Land 220, 261, 262, 308 Mechanics 276 Metallurgy 244, 253, 291, 292 Monuments 289 Paint 290, 291, 308 Parks and recreation 244, 260, 261, 291, 292, 308 Satellites 262 Science 210, 211, 220, 221, 262, 287, 288 Sports and fitness 210, 215, 220, 221, 245, 246, 250, 259, 260, 261, 262, 263, 290, 301, 308 Section 4.1 The Metric System 203 Investigation: Electrical Energy 207 Section 4.2 The U.S. Customary System 211 Investigation: Energy 218 Section 4.3 Basic Concepts of Euclidean Geometry 221 Investigation: Preparing a Circle Graph 234 Section 4.4 Perimeter and Area of Plane Figures 241 Investigation: Slicing Polygons into Triangles 257 Section 4.5 Properties of Triangles 265 Investigation: Topology 271 CHAPTER 4 Measurement and Geometric Models 202 Contents v APPLICATIONS Aviation 327, 348 Business 334, 339, 344, 345, 364, 365, 366 Compensation 345 Construc- tion 336, 345 Education 346, 362 Farming 367 Forestry 322 Geometry 318, 321, 322, 323, 337, 364 Health 311, 335, 340, 341, 346, 358, 359, 361, 365, 367 Internet 366 Money 334, 335, 342, 346, 361, 366 Popula- tions 367 Recreation 321, 324, 327, 330, 341, 344, 348, 367 Science 322, 323, 332, 334, 336, 339, 347, 351, 352, 355, 359, 364, 365 Sports 322, 323, 335, 351 Telecommunications 361, 365 Temperature 330, 361 Vehicles 338, 347, 360, 362 Section 5.1 Rectangular Coordinates and Functions 312 Investigation: Dilations of a Geometric Figure 320 Section 5.2 Properties of Linear Functions 324 Investigation: Negative Velocity 332 Section 5.3 Finding Linear Models 337 Investigation: A Linear Business Model 344 Section 5.4 Linear Regression and Correlation 349 Investigation: An Application of Linear Regression 355 Chapter Summary 363 • Review Exercises 364 • Chapter Test 367 CHAPTER 5 Linear Models 311 APPLICATIONS Astronomy 410, 413 Automotive technology 381, 396 Business 377, 380, 382, 383, 412 Chemistry 409, 412, 414 Construction 382 Earth science 396, 409, 410, 414 Engineering 380 Finance 396, 413 Fountains 381 Geometry 375, 381, 382, 412, 414 Hot-air balloon 382 Manufacturing 380, 413 Meteorology 397 Music 396 Oil supply 410 Physics 380, 383, 396, 397, 412, 413, 414 Population 396, 397 Ranching 381 Recreation 381, 382 Space science 382 Sports 380, 381, 412, 413 Stopping distance 381 Sustainable yield 383 Water treatment 380 Welding 396 Section 6.1 Introduction to Nonlinear Functions 369 Investigation: Reflective Properties of a Parabola 378 Section 6.2 Exponential Functions 386 Investigation: Chess and Exponential Functions 394 Section 6.3 Logarithmic Functions 398 Investigation: Benford’s Law 407 Chapter Summary 411 • Review Exercises 412 • Chapter Test 414 CHAPTER 6 Nonlinear Models 368 Storage 260, 282, 291 Surveying 275, 309, 310 Travel 276, 301, 302 Trigonometry see Section 4.7 Section 4.6 Volume and Surface Area 277 Investigation: Water Displacement 286 Section 4.7 Introduction to Trigonometry 292 Investigation: Approximating the Values of Trigonometric Functions 299 Chapter Summary 305 • Review Exercises 307 • Chapter Test 309 vi Contents APPLICATIONS Annual yield 448, 491, 493 Bonds 471, 474, 476, 491, 492 Buying on credit 451, 454, 455, 461, 462, 463, 465, 491, 493 Car leases and purchases 456, 459, 461, 463, 464, 466, 491, 492 Compound interest 433, 434, 435, 445, 446, 448, 491, 492 Cost of living 415, 444 Effective interest rate 441, 447, 491, 493 Future value 421, 426, 430, 434, 445, 446, 448, 491 Home ownership 477, 484, 485, 486, 487, 488, 492, 493 Inflation 438, 439, 447, 491, 493 Loans 424, 458, 463, 464, 491, 493 Maturity value 420, 421, 426, 427, 490, 492 Mortgages 479, 481, 483, 486, 487, 488, 492, 493 Present value 437, 445, 446, 448, 491, 492 Simple interest 416, 417, 418, 422, 426, 427, 428, 490, 492 Stock market 467, 469, 472, 474, 475, 476, 492, 493 Section 7.1 Simple Interest 416 Investigation: Day-of-the-Year Table 423 Section 7.2 Compound Interest 429 Investigation: Consumer Price Index 443 Section 7.3 Credit Cards and Consumer Loans 449 Investigation: Leasing Versus Buying a Car 460 Section 7.4 Stocks, Bonds, and Mutual Funds 467 Investigation: Treasury Bills 473 Section 7.5 Home Ownership 477 Investigation: Home Ownership Issues 485 Chapter Summary 489 • Review Exercises 490 • Chapter Test 492 CHAPTER The Mathematics of Finance 415 7 APPLICATIONS Academy awards 536, 537, 549, 561, 579 Automotive industry 505, 578, 590, 592 Banking 504, 579 Business 504, 537, 577, 592 Communication technology 578 Commuting 539, 562 Computers 506 Construction 505 Demo- graphics 523 Dice 495, 510, 526 Drug testing 519, 525 Education 505, 506, 524, 537, 561, 562 Elec- tions 592, 593 Entertainment 506 Environment 592 Genetics 506, 520, 525 Geometry 504 Health science 525, 561, 562, 590, 591, 592 Height distributions 578, 579 Housing 562, 563, 592 Human resources 505, 524 Incomes 562, 577 Internet 506 Library science 505 Lotteries 525, 526, 549, 590 Meteorology 538, 548 Monty Hall problem 526 Physical fitness 561 Playing cards 502, 510, 513, 523, 524 Political science 550 Quality control 505, 591, 592 Recreation 523, 549 Restaur- ants 505, 590 Roulette 524, 591 Salaries 523, 563, 577 Sports 499, 500, 505, 506, 539, 540, 549, 550, 562, 563, 578, 591, 592 Test- ing 505, 538, 539, 540, 561, 562, 577, 579 Weight distributions 549, 578, 579, 592 Section 8.1 Counting Methods 495 Investigation: Choosing Numbers in Keno 502 Section 8.2 Introduction to Probability 507 Investigation: Complement of an Event and Sharing Birthdays 521 Section 8.3 Measures of Central Tendency 527 Investigation: Linear Interpolation and Animation 535 Section 8.4 Measures of Dispersion 540 Investigation: A Geometric View of Variance and Standard Deviation 547 Section 8.5 Measures of Relative Position 552 Investigation: Stem-and-Leaf Diagrams 558 Section 8.6 Normal Distributions 564 Investigation: Cut-Off Scores 576 Section 8.7 Inferential Statistics 580 Investigation: A Fair Die? 589 Chapter Summary 593 • Review Exercises 595 • Chapter Test 600 CHAPTER Probability and Statistics 494 8 Solutions to Check Your Progress Problems S1 Answers to Selected Exercises A1 Index I1 Web Appendix: Algebra Review (Available only online at this textbook’s Online Study Center at: hmco.college.com/pic/aufmannMTQR.) Contents vii APPLICATIONS Architecture 677, 678, 681, 683, 684, 719 Bicycling 675 Computer networking 695, 701, 721 Machine configuration 694 Map coloring 702, 703, 704, 705, 712, 715, 716, 721, 724 Overnight delivery 676 Parks 682, 683, 719 Pen-tracing puzzles 678 Pets 683 Route planning 684, 694, 698, 700, 702, 721 Scheduling 700, 707, 708, 709, 714, 715, 716, 722, 724 Social networks 681, 722 Sports 681, 716, 718 Traffic signals 710 Travel 673, 674, 676, 680, 683, 684, 685, 686, 691, 692, 697, 698, 699, 718, 720, 723 World Wide Web 669 Section 10.1 Traveling Roads and Visiting Cities 667 Investigation: Pen-Tracing Puzzles 678 Section 10.2 Efficient Routes 685 Investigation: Extending the Greedy Algorithm 696 Section 10.3 Map Coloring and Graphs 702 Investigation: Modeling Traffic Lights with Graphs 710 Chapter Summary 717 • Review Exercises 718 • Chapter Test 722 CHAPTER The Mathematics of Graphs 666 APPLICATIONS Apportionment principle 619 Business 660, 661, 662, 665 Cell phones 639, 640 Criminal justice 656 Computers 621, 622, 641 Construction 661 Demographics 622 Education 617, 618, 620, 621, 639, 640, 642, 660, 662, 663, 664, 665 Elections 637, 642, 643, 644, 658 Entertainment 638, 639, 641, 643 Family reunion 641 Food science 637, 638 Government 618, 621, 664 Health 619 Manage- ment 619, 620, Music 656, 661 Radio stations 640 Recreation 641 Restaurants 638, 639, 645 Scholarships 644, 662 Shapley- Shubik Power Index 657 Social science 621 Sports 639, 657, 663 Technology 661 Transportation 661 Section 9.1 Introduction to Apportionment 603 Investigation: Apportioning the 1790 House of Representatives 615 Section 9.2 Introduction to Voting 623 Investigation: Variations of the Borda Count Method 636 Section 9.3 Weighted Voting Systems 646 Investigation: Blocking Coalitions and the Banzhaf Power Index 654 Chapter Summary 659 • Review Exercises 660 • Chapter Test 664 CHAPTER Apportionment and Voting 602 9 10 M athematical Thinking and Quantitative Reasoning presents an analytical investigation of topics and concepts that are relevant to modern society. Our goal is to demonstrate the power of mathematics and quantitative reasoning in solving contemporary problems. Mathematical Thinking and Quantitative Reasoning provides glimpses into how mathematics is used to solve real-life problems. Students will learn how prime num- bers are used to encrypt information sent across the Internet, the role of modular arithmetic in verifying credit card numbers, how to determine whether to lease or buy a car, how statistics is used to predict the outcome of elections, and how math- ematics can be used to evaluate voting systems. Two features that we have incorporated in the text are Math Matters and Investiga- tions. Math Matters are vignettes of interesting applications related to the topic being discussed. Each section of the text ends with an Investigation, which is an extension of one of the topics presented in that section. For instance, one Investiga- tion extends the ideas of formal logic to logic gates in computers; another Investigation examines how to determine whether a die is fair. The exercise sets in Mathematical Thinking and Quantitative Reasoning have been carefully selected to reinforce and extend the concepts developed in each section. The exercises range from drill-and-practice to interesting challenges. Some of the exercise sets include outlines for further explorations, suggestions for essays, critical thinking problems, and cooperative learning activities. In all cases, the exercises were chosen to illustrate the many facets of the topic under discussion. The purpose of this text is to strengthen students’ quantitative reasoning skills by having them solve a variety of real-world problems. Although we assume that the reader has an intermediate algebra background, each topic is carefully developed, and appropriate material is reviewed whenever necessary. When deciding on the depth of coverage, our singular criterion was to make mathematics accessible. viii P R E F A C E ix Chapter 1 • Problem Solving An Interactive Method Mathematical Thinking and Quantitative Reasoning is written in a style that encourages the student to interact with the textbook. Each section contains a variety of worked examples. Each example is given a title so that the student can see at a glance the type of problem that is being solved. Most examples include annotations that assist the student in moving from step to step, and the final answer is in color in order to be readily identifiable. Check Your Progress Exercises Following each worked example is a Check Your Progress exercise for the student to work. By solving this exercise, the student actively practices concepts as they are presented in the text. For each Check Your Progress exercise, there is a detailed solution in the Solutions appendix. page 128 128 Chapter 3 • Algebraic Models Humerus Applications In some applications of equations, we are given an equation that can be used to solve the application. This is illustrated in Example 3. EXAMPLE 3 ■ Solve an Application Forensic scientists have determined that the equation can be used to approximate the height H, in centimeters, of an adult on the basis of the length L, in centimeters, of the adult’s humerus (the bone extending from the shoulder to the elbow). a. Use this equation to approximate the height of an adult whose humerus meas- ures 36 centimeters. b. According to this equation, what is the length of the humerus of an adult whose height is 168 centimeters? Solution a. Substitute 36 for L in the given equation. Solve the resulting equation for H. The adult’s height is approximately 182.5 centimeters. b. Substitute 168 for H in the given equation. Solve the resulting equation for L. The length of the adult’s humerus is approximately 31 centimeters. CHECK YOUR PROGRESS 3 The amount of garbage generated by each person living in the United States has been increasing and is approximated by the equation where P is the number of pounds of garbage generated per per- son per day and Y is the year. a. Find the amount of garbage generated per person per day in 1990. b. According to the equation, in what year will 5.6 pounds of garbage be generated per person per day? Solution See page S11. P � 0.05Y � 95, 31 � L 89.9 2.9 � 2.9L 2.9 89.9 � 2.9L 168 � 78.1 � 2.9L � 78.1 � 78.1 168 � 2.9L � 78.1 H � 2.9L � 78.1 H � 182.5 H � 104.4 � 78.1 H � 2.9�36� � 78.1 H � 2.9L � 78.1 H � 2.9L � 78.1 Preface ix W elcome to Mathematical Thinking and Quantitative Reasoning. As you begin this course, we know two important facts: (1) You want to succeed. (2) We want you to succeed. In order to accomplish these goals, an effort is required from each of us. For the next few pages, we are going to show you what is required of you to achieve your goal and how we have designed this text to help you succeed. Motivation One of the most important keys to success is motivation.We can try to motivate you by offering interesting or important ways that you can benefit from mathematics. But, in the end, the motivation must come from you. On the first day of class it is easy to be motivated. Eight weeks into the term, it is harder to keep that motivation. To stay motivated, there must be outcomes from this course that are worth your time, money, and energy. List some reasons you are taking this course. Do not make a mental list—actually write them out. Do this now. Although we hope that one of the reasons you listed was an interest in mathemat- ics, we know that many of you are taking this course because it is required to gradu- ate.Although you may not agree that this course should be necessary, it is! If you are motivated to graduate or complete the requirements for your major, then use that motivation to succeed in this course. Do not become distracted from your goal of completing your education! Commitment To be successful, you must make a commitment to succeed. This means devoting time to math so that you achieve a better understanding of the subject. List some activities (sports, hobbies, talents such as dance, art, or music) that you enjoy and at which you would like to become better. Do this now. Next to these activities, put the number of hours each week that you spend practic- ing these activities. Whether you listed surfing or sailing, aerobics or restoring cars, or any other activ- ity you enjoy, note how many hours a week you spend on each activity. To succeed in math, you must be willing to commit the same amount of time. Success requires some sacrifice. xvii ✔ TAKE NOTE Motivation alone will not lead to success. For instance, suppose a person who cannot swim is placed in a boat, taken out to the middle of a lake, and then thrown overboard. That person has a lot of motivation to swim, but there is a high likelihood the person will drown without some help. Motivation gives us the de- sire to learn but is not the same as learning. A I M F O R S U C C E S S AIM for Success Student Preface This “how to use this text” preface explains what is re- quired of a student to be successful and how this text has been designed to foster student success. AIM for Success can be used as a lesson on the first day of class or as a project for students to complete to strengthen their study skills. page xvii Mathematical Thinking and Quantitative Reasoning is designed to foster student success through an inte- grated text and media program. Student Success x Chapter 1 • Problem Solving 1.1 • Inductive and Deductive Reasoning x Investigations Each section ends with an Investigation along with corresponding Investigation Exercises. These activities engage students in the mathematics of the section. Some Investigations are designed as in-class cooperative learning activities that lend themselves to a hands-on approach. They can also be assigned as projects or extra credit assignments. The Investigations are a unique and important feature of this text. They provide opportunities for students to take an active role in the learning process. 3.2 • Rate, Ratio, and Proportion 151 Investigation Earned Run Average One measure of a pitcher’s success is earned run average. Earned run average (ERA) is the number of earned runs a pitcher gives up for every nine innings pitched. The defi- nition of an earned run is somewhat complicated, but basically an earned run is a run that is scored as a result of hits and base running that involves no errors on the part of the pitcher’s team. If the opposing team scores a run on an error (for example, a fly ball that should be caught in the outfield is fumbled), then that run is not an earned run. A proportion is used to calculate a pitcher’s ERA. Remember that the statistic involves the number of earned runs per nine innings. The answer is always rounded to the near- est hundredth. Here is an example. During the 2005 baseball season, Pedro Martinez gave up 68 earned runs and pitched 217 innings for the New York Mets. To calculate Pedro Martinez’s ERA, let x equal the number of earned runs for every nine innings pitched. Write a proportion and then solve it for x. Pedro Martinez’s ERA for the 2005 season was 2.82. Investigation Exercises 1. In 1979, his rookie year, Jeff Reardon pitched 21 innings for the New York Mets and gave up four earned runs. Calculate Reardon’s ERA for 1979. 2. Roger Clemens’s first year with the Boston Red Sox was 1984. During that season, he pitched 133.1 innings and gave up 64 earned runs. Calculate Clemens’s ERA for 1984. 3. During the 2003 baseball season, Ben Sheets of the Milwaukee Brewers pitched 220.2 innings and gave up 109 earned runs. During the 2004 season, he gave up 71 earned runs and pitched 237.0 innings. During which season was his ERA lower? How much lower? 4. In 1987, Nolan Ryan had the lowest ERA of any pitcher in the major leagues. He gave up 65 earned runs and pitched 211.2 innings for the Houston Astros. Calculate Ryan’s ERA for 1987. 5. Find the necessary statistics for a pitcher on your “home team,” and calculate that pitcher’s ERA. 2.82 � x 612 217 � 217x 217 612 � 217x 68 � 9 � 217 � x 68 earned runs 217 innings � x 9 innings National League Earned Run Average Leaders Player, club Year ERA 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Danny Darwin, Houston Dennis Martinez, Montreal Bill Swift, San Francisco Greg Maddux, Atlanta Greg Maddux, Atlanta Greg Maddux, Atlanta Kevin Brown, Florida Pedro Martinez, Montreal Greg Maddux, Atlanta Randy Johnson, Arizona Kevin Brown, Los Angeles Randy Johnson, Arizona Randy Johnson, Arizona Jason Schmidt, San Francisco Jake Peavy, San Diego Roger Clemens, Houston 2.21 2.39 2.08 2.36 1.56 1.63 1.89 1.90 2.22 2.48 2.58 2.49 2.32 2.34 2.27 1.87 American League 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Roger Clemens, Boston Roger Clemens, Boston Roger Clemens, Boston Kevin Appier, Kansas City Steve Ontiveros, Oakland Randy Johnson, Seattle Juan Guzman, Toronto Roger Clemens, Toronto Roger Clemens, Toronto Pedro Martinez, Boston Pedro Martinez, Boston Freddy Garcia, Seattle Pedro Martinez, Boston Pedro Martinez, Boston Johan Santana, Minnesota Kevin Millwood, Cleveland 1.93 2.62 2.41 2.56 2.65 2.48 2.93 2.05 2.65 2.07 1.74 3.05 2.26 2.22 2.61 2.86 Student Success, continued page 151 x Preface 4.1 • The Metric System 203 SECTION 4.1 The Metric System The Metric System International trade, or trade between nations, is a vital and growing segment of business in the world today. The United States, as a nation, is dependent on world trade, and world trade is dependent on internationally standardized units of meas- urement: the metric system. In this section we will present the metric system of measurement and explain how to convert between different units. The basic unit of length, or distance, in the metric system is the meter (m). One meter is approximately the distance from a doorknob to the floor. All units of length in the metric system are derived from the meter. Prefixes to the basic unit denote the length of each unit. For example, the prefix “centi-” means one-hundredth; there- fore, one centimeter is 1 one-hundredth of a meter (0.01 m). kilo- � 1 000 1 kilometer (km) � 1 000 meters (m) hecto- � 100 1 hectometer (hm) � 100 m deca- � 10 1 decameter (dam) � 10 m 1 meter (m) � 1 m deci- � 0.1 1 decimeter (dm) � 0.1 m centi- � 0.01 1 centimeter (cm) � 0.01 m milli- � 0.001 1 millimeter (mm) � 0.001 m Note that in this list 1000 is written as 1 000, with a space between the 1 and the zeros.When writing numbers using metric units, separate each group of three num- bers by a space instead of a comma. A space is also used after each group of three numbers to the right of a decimal point. For example, 31,245.2976 is written 31 245.297 6 in metric notation. Which unit in the metric system is one-thousandth of a meter? Mass and weight are closely related. Weight is the force of gravity pulling on an object. Therefore, an object’s weight is less in space than on Earth’s surface. However, the amount of material in the object, its mass, remains the same. On the surface of Earth, the terms mass and weight are often used interchangeably. The basic unit of mass in the metric system is the gram (g). If a box 1 cen- timeter long on each side is filled with water, the mass of that water is 1 gram. QUESTION The millimeter is one-thousandth of a meter. ANSWER point of interest Originally the meter (spelled in some countries) was defined as of the distance from the equator to the North Pole. Modern scientists have redefined the meter as the distance light travels in a vacuum in second. 1 1,299,792,458 1 1,000,000 metre ≈1 meter 1 gram � the mass of water in a box that is 1 centimeter long on each side 1 cm 1 cm 1 cm page 203 Question/Answer Feature At various places throughout the text, a Question is posed about the topic that is being developed. This question encourages students to pause, think about the current discussion, and answer the question. Stu- dents can immediately check their understanding by referring to the Answer to the question provided in a footnote on the same page. This feature creates an- other opportunity for the student to interact with the textbook. 1.1 • Inductive and Deductive Reasoning xi xi Chapter 1 • Problem Solving Math Matters This feature of the text typically contains an interesting sidelight about mathematics, its history, or its applications. Conceptual Understanding Mathematical Thinking and Quantitative Reasoning helps students understand the course concepts through features in the exposition. Calculator Note These notes provide information about how to use the various features of a calculator. 372 Chapter 6 • Nonlinear Models EXAMPLE 2 ■ Find the Vertex of a Parabola Find the vertex of the parabola whose equation is Solution • Find the x-coordinate of the vertex. • Find the y-coordinate of the vertex by replacing x by 1 and solving for y. The vertex is CHECK YOUR PROGRESS 2 Find the vertex of the parabola whose equation is Solution See page S22. MathMatters Paraboloids The movie Contact was based on a novel by astronomer Carl Sagan. In the movie, Jodie Foster plays an astronomer who is searching for extraterrestrial intelligence. One scene from the movie takes place at the Very Large Array (VLA) in New Mex- ico. The VLA consists of 27 large radio telescopes whose dishes are paraboloids, the three-dimensional version of a parabola. A parabolic shape is used because of the following reflective property: When parallel rays of light, or radio waves, strike the surface of a parabolic mirror whose axis of symmetry is parallel to these rays, they are reflected to the same point. The photos above show the layout of the radio telescopes of the VLA and a more detailed picture of one of the telescopes. The figure at the far right shows the reflective property of a parabola. All the incoming rays are reflected to a point called the focus. The reflective property of a parabola is also used in optical telescopes and head- lights on a car. In the case of headlights, the bulb is placed at the focus and the light is reflected along parallel rays from the reflective surface of the headlight, thereby making a more concentrated beam of light. Incoming signal Axis (parallel to incoming signal) Receiver located at the focus Parabolic satellite dish Center y � x 2 � 2. �1, 4�. y � 4 y � �3�1�2 � 6�1� � 1 y � �3x 2 � 6x � 1 a � �3, b � 6 x � � b 2a � � 6 2��3� � 1 y � �3x 2 � 6x � 1. Historical Note These margin notes provide historical background in- formation related to the concept under discussion or vignettes of individuals who were responsible for major advancements in their fields of expertise. page 372 page 247 Take Note These notes alert students to a point requiring special attention or are used to amplify the concepts that are currently being developed. Point of Interest These notes provide interesting information related to the topics under discussion. Many of these are of a contemporary nature and, as such, they provide stu- dents with the needed motivation for studying con- cepts that may at first seem abstract and obscure. A circle is a plane figure in which all points are the same distance from point O, called the center of the circle. A diameter of a circle is a line segment with endpoints on the circle and passing through the center. is a diameter of the circle at the right. The variable d is used to designate the length of a diameter of a circle. A radius of a circle is a line segment from the center of the circle to a point on the circle. is a radius of the circle at the right above. The variable r is used to designate the length of a radius of a circle. The length of the diameter is twice the length of the radius. The distance around a circle is called the cir- cumference. The formula for the circumfer- ence, C, of a circle is: Because the formula for the circum- ference can also be written as: Circumference of a Circle The circumference C of a circle with diameter d and radius r is given by or The formula for circumference uses the num- ber (pi), which is an irrational number. The value of can be approximated by a fraction or by a decimal. The key on a scientific calculator gives a closer approximation of than 3.14. A scientific calculator is used in this section to find approximate values in calcula- tions involving Find the circumference of a circle with a diameter of 6 m. The diameter of the circle is given. Use the circumference formula that involves the diameter. The exact circumference of the circle is An approximate measure can be found by using the key on a calculator. An approximate circumference is 18.85 m. � 6� m. d � 6. �. � � � � C � 2�r. C � �d d � 2r, OC AB 4.4 • Perimeter and Area of Plane Figures 247 A B C O or C � 2�r C � �d r � 1 2 d d � 2r point of interest A glazier is a person who cuts, fits, and installs glass, generally in doors and windows. Of particular challenge to a glazier are intricate stained glass window designs. or � � 3.14 � � 3 1 7 C � 18.85 C � 6� C � ��6� C � �d ✔ TAKE NOTE Recall that an irrational number is a number whose decimal representation never terminates and does not have a pattern of numerals that keep repeating. CALCULATOR NOTE The key on your calculator can be used to find decimal approxi- mations of expressions that con- tain . To perform the calculation at the right, enter 6 . � � � � � Preface xi xii Chapter 1 • Problem Solving 1.1 • Inductive and Deductive Reasoning xii Exercise Sets The exercise sets were carefully written to provide a wide variety of exercises that range from drill- and-practice to interesting challenges. Exercise sets emphasize skill building, skill maintenance, concepts, and applications. Icons are used to identify various types of exercise. Conceptual Understanding, continued 152 Chapter 3 • Algebraic Models Exercise Set 3.2 1. Provide two examples of situations in which unit rates are used. 2. Explain why unit rates are used to describe situ- ations involving units such as miles per gallon. 3. What is the purpose of exchange rates in inter- national trade? 4. Provide two examples of situations in which ratios are used. 5. Explain why ratios are used to describe situations involving information such as student–teacher ratios. 6. What does the phrase “the cross products are equal” mean? 7. Explain why the product of the means in a pro- portion equals the product of the extremes. In Exercises 8–13, write the expression as a unit rate. 8. 582 miles in 12 hours 9. 138 miles on 6 gallons of gasoline 10. 544 words typed in 8 minutes 11. 100 meters in 8 seconds 12. $9100 for 350 shares of stock 13. 1000 square feet of wall covered with 2.5 gallons of paint 14. A rate of 288 miles in 6 hours is closest to which unite rate? a. 100 mph b. 50 mph c. 200 mph d. 25 mph 15. A rate of $123.75 in 11 hours is closest to which unite rate? a. $20 per hour b. $1 per hour c. $10 per hour d. $120 per hour Solve Exercises 16–21. 16. Wages A machinist earns $490 for working a 35-hour week. What is the machinist’s hourly rate of pay? 17. Space Vehicles Each of the Space Shuttle’s solid rocket motors burns 680,400 kilograms of propellant in 2.5 minutes. How much propellant does each motor burn in 1 minute? 18. Photography During filming, an IMAX camera uses 65-mm film at a rate of 5.6 feet per second. a. At what rate per minute does the camera go through film? b. How quickly does the camera use a 500-foot roll of 65-mm film? Round to the nearest second. 19. Consumerism Which is the more economical purchase, a 32-ounce jar of may- onnaise for $2.79 or a 24-ounce jar of mayonnaise for $2.09? 20. Consumerism Which is the more economical purchase, an 18-ounce box of corn flakes for $2.89 or a 24-ounce box of corn flakes for $3.89? 21. Wages You have a choice of receiving a wage of $34,000 per year, $2840 per month, $650 per week, or $16.50 per hour. Which pay choice would you take? Assume a 40-hour work week and 52 weeks of work per year. 22. Baseball Baseball statisticians calculate a hitter’s at-bats per home run by dividing the number of times the player has been at bat by the number of home runs the player has hit. a. Calculate the at-bats per home run ratio for each player in the table on the following page. Round to the nearest tenth. b. Which player has the lowest rate of at-bats per home run? Which player has the second lowest rate? Babe Ruth 684 Chapter 10 • The Mathematics of Graphs 41. Transportation For the train routes given in Exercise 5, find a route that visits each city and returns to the starting city without visiting any city twice. 42. Transportation For the direct air flights given in Ex- ercise 6, find a route that visits each city and returns to the starting city without visiting any city twice. 43. Architecture In Exercise 7, you were asked to draw a graph that represents a museum floor plan. Describe what a Hamiltonian circuit in the graph would corre- spond to in the museum. 44. Transportation Consider a subway map, like the one given in Exercise 32. If we draw a graph in which each vertex represents a train junction, and an edge between vertices means that a train travels between those two junctions, what does a Hamiltonian circuit correspond to in regard to the subway? Extensions C R IT I C A L T H I N K I NG 45. Route Planning A security officer patrolling a city neighborhood needs to drive every street each night. The officer has drawn a graph representing the neighborhood, in which the edges represent the streets and the vertices correspond to street intersec- tions. Would the most efficient way to drive the streets correspond to an Euler circuit, a Hamiltonian circuit, or neither? (The officer must return to the starting lo- cation when finished.) Explain your answer. 46. Route Planning A city engineer needs to in- spect the traffic signs at each street intersection of a neighborhood. The engineer has drawn a graph representing the neighborhood, where the edges rep- resent the streets and the vertices correspond to street intersections. Would the most efficient route corre- spond to an Euler circuit, a Hamiltonian circuit, or neither? (The engineer must return to the starting lo- cation when finished.) Explain your answer. 47. Is there an Euler circuit in the graph below? Is there an Euler walk? Is there a Hamiltonian circuit? Justify your answer. (You do not need to find any of the cir- cuits or paths.) 48. Is there an Euler circuit in the graph below? Is there an Euler walk? Is there a Hamiltonian circuit? Justify your answer. (You do not need to find any of the cir- cuits or paths.) CO O PERATI VE LEA RN I N G 49. a. Draw a connected graph with six vertices that has no Euler circuit and no Hamiltonian circuit. b. Draw a graph with six vertices that has a Hamil- tonian circuit but no Euler circuit. c. Draw a graph with five vertices that has an Euler circuit but no Hamiltonian circuit. 50. Travel A map of South America is shown below. a. Draw a graph in which the vertices represent the 13 countries of South America, and two vertices are joined by an edge if the corresponding coun- tries share a common border. b. Two friends are planning a driving tour of South America. They would like to drive across every border on the continent. Is it possible to plan such a route that never crosses the same border twice? What would the route correspond to on the graph? c. Find a route the friends can follow that will start and end in Venezuela and that crosses every border while recrossing the fewest borders possible. Hint: On the graph, add multiple edges corresponding to border crossings that allow an Euler circuit. Brazil Bolivia Paraguay Uruguay Chile Ecuador Colombia Guyana Suriname French Guiana Venezuela Ar ge nt in a P e r u Extensions Extension exercises are placed at the end of each exercise set. These exercises are designed to extend concepts. In most cases these exercises are more chal- lenging and require more time and effort than the preceding exercises. The Extension exercises always include at least two of the following types of exercises: Critical Thinking Cooperative Learning Explorations Some Critical Thinking exercises require the applica- tion of two or more procedures or concepts. The Cooperative Learning exercises are designed for small groups of two to four students. Many of the Exploration exercises require students to search for information on the Internet or through reference materials in a library. Writing exercises Data analysis exercises Graphing calculator exercises Internet exercises Think About It exercises page 152 page 684 xii Preface The Think About It exercises are conceptual in nature. They ask students to think about the concepts studied in a sec- tion, make generalizations, and apply them to more abstract problems. The focus is on men- tal mathematics, not calculation or computation. These exercises are meant to help students synthesize concepts. Mathematical Thinking and Quantitative Reasoning helps students understand course concepts through features in the exercise sets. 1.1 • Inductive and Deductive Reasoning xiii xiii Chapter 1 • Problem Solving Chapter Summary At the end of each chapter there is a Chapter Summary that includes Key Terms and Essential Concepts that were covered in the chapter. These chapter summaries provide a single point of reference as the student pre- pares for an examination. Each key word references the page number where the word was first introduced. Assessment and Review Chapter 5 • Summary 363 Key Terms abscissa [p. 313] coordinate axes [p. 312] coordinates of a point [p. 313] dependent variable [p. 317] domain [p. 316] equation in two variables [p. 313] evaluating a function [p. 317] extrapolation [p. 352] first coordinate [p. 313] function [p. 316] function notation [p. 317] graph a point [p. 313] graph of an equation in two variables [p. 314] graph of an ordered pair [p. 313] independent variable [p. 317] interpolation [p. 352] linear correlation coefficient [p. 353] linear function [p. 324] linear regression [p. 349] negative correlation [p. 353] ordered pair [p. 313] ordinate [p. 313] origin [p. 312] plot a point [p. 313] positive correlation [p. 353] quadrants [p. 312] range [p. 316] rectangular coordinate system [p. 312] regression line [p. 349] scatter diagram [p. 341] second coordinate [p. 313] slope [p. 328] solution of an equation in two variables [p. 313] value of a function [p. 317] x-coordinate [p. 313] x-intercept [p. 325] xy-plane [p. 312] y-coordinate [p. 313] y-intercept [p. 325] CHAPTER 5 Summary 364 Chapter 5 • Linear Models 1. Draw a line through all points with an x-coordinate of 4. 2. Draw a line through all points with a y-coordinate of 3. Graph the ordered-pair solutions of when 4. Graph the ordered-pair solutions of when 5. Graph the ordered-pair solutions of when 6. Graph the ordered-pair solutions of when In Exercises 7–12, graph the function. 7. 8. 9. 10. 11. 12. In Exercises 13–16, evaluate the function for the given value. 13. In Exercises 20–23, find the x- and y-intercepts of the graph of the function. 20. 21. 22. 23. 24. Depreciation The accountant for a small busi- ness uses the model to approximate the value of a small truck t years after its purchase. Find and discuss the meaning of the intercepts, on the vertical and horizontal axes, in the context of this application. In Exercises 25–28, find the slope of the line passing through the given points. 25. 26. 27. 28. 29. The Film Industry The following graph shows annual projections for revenue from home video rentals. Find the slope of the line and write a sentence that explains the meaning of the slope in the f h l ( h) �5, 2�, �5, 7� �2, �5�, ��4, �5� ��1, 4�, ��3, �1� �3, 2�, �2, �3� V�t� V�t� � 25,000 � 5000t 4x � 3y � 24 3x � 5y � 15 f�x� � 3 4 x � 9 f�x� � 2x � 10 f(x) � 3x � 5; x � �3 f�x� � 2�x� � 1 y � �x � 4� f�x� � x 2 � 3x � 1 f�x� � x 2 � 2 y � 3x � 2 y � �2x � 1 x � �5, �3, 0, 3, and 5. y � �x � 1� x � �2, �1, 0, 1, and 2. y � �2x � 1 x � �2, �1, 0, 1, and 2. y � 2x 2 � 5 x � �2, �1, 0, 1, and 2. y � 2x 2 3. CHAPTER 5 Review Exercises Chapter 5 • Test 367 In Exercises 1 and 2, evaluate the function for the given value of the independent variable. 1. 2. t � �2 f�(x) � �x� � 2; x � �3 s�t� � �3t 2 � 4t � 1; In Exercises 3 and 4, graph the function. 3. 4. 5. Find the and intercepts of the line given by 6. Find the slope of the line that passes through and 7. Find the equation of the line that passes through and has slope 8. Travel The distance d, in miles, a small plane is from its final destination is given by where t is the time, in hours, re- maining for the flight. Find and discuss the meaning of the intercepts of the graph of the function. d�t� � 250 � 100t, 2 3. �3, 5� ��2, �4�. �3, �1� 5x � 3y � 8. y- x- f�x� � x 2 � 2x � 3 f�x� � 2x � 3 11. Population The following table shows the estimated population, in thousands, of England for various years. Source: UK Office for National Statistics, www.statistics.gov.uk Find the equation of the regression line and the linear regression coefficient for the data.According to the re- gression coefficient, how well does the regression line fit the data? 12 N t iti Th f ll i bl h h Year Population (thousands) 1994 48,229 2004 50,094 2011 51,967 2021 54,605 2031 56,832 CHAPTER 5 Test Chapter Review Exercises Review Exercises are found near the end of each chap- ter. These exercises were selected to help the student integrate the major topics presented in the chapter. The answers to all Review Exercises appear in the an- swer section, along with a section reference for each exercise. These section references indicate the section or sections in which a student can locate the concepts needed to solve each exercise. Chapter Test The Chapter Test exercises are designed to emulate a possible test of the material in the chapter. The answers to all Chapter Test exercises appear in the answer sec- tion, along with a section reference for each exercise. The section references indicate the section or sections in which a student can locate the concepts needed to solve each exercise. page 363 page 364 page 367 Preface xiii Mathematical Thinking and Quantitative Reasoning is designed to foster student success through practice and review. Supplements for the Instructor Mathematical Thinking and Quantitative Reasoning has an extensive support pack- age for the instructor that includes: Instructor’s Annotated Edition (IAE): The Instructor’s Annotated Edition is an exact replica of the student textbook with the following additional text-specific items for the instructor: answers to all of the end-of-section and end-of-chapter exercises, answers to all Investigation and Exploration exercises, Instructor Notes, Suggested Assignments, and icons denoting tables and art that appear in PowerPoint® slides. Online Teaching Center: This free companion website contains an abundance of in- structor resources such as solutions to all exercises in the text, digital art and tables, suggested course syllabi, Chapter Tests, a Graphing Calculator Guide, and Microsoft® Excel spreadsheets. Visit hmco.college.com/pic/aufmannMTQR and click on the Online Teaching Center icon. Online Instructor’s Solutions Manual: The Online Instructor’s Solutions Manual offers worked-out solutions to all of the exercises in each exercise set as well as solu- tions to the Investigation and Exploration exercises. HM Testing CD-ROM (powered by Diploma™): HM Testing (powered by Diploma) offers instructors a flexible and powerful tool for test generation and test management. Now supported by the Brownstone Research Group’s market-leading Diploma software, this new version of HM Testing significantly improves on func- tionality and ease of use by offering all the tools needed to create, author, deliver, and customize multiple types of tests—including authoring and editing algorithmic questions. Eduspace®: Eduspace, powered by Blackboard®, is Houghton Mifflin’s customiz- able and interactive online learning tool. Eduspace provides instructors with online courses and content. By pairing the widely recognized tools of Blackboard with quality, text-specific content from Houghton Mifflin Company, Eduspace makes it easy for instructors to create all or part of a course online. This online learning tool also contains ready-to-use homework exercises, quizzes, tests, tutorials, and supplemental study materials. Visit eduspace.com for more information. Preface xiv
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