Algebra for the Sciences Textbook

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Algebra for the Sciences

K.Oty B.Elliott

August 6, 2018 i Preface

This book was developed as an alternative to the traditional college algebra book currently used at many universities. The first edition was written while we were partially funded by a grant from the National Science Foundation and some of the projects in the second edition were written with the support of the Ok- lahoma IDeA Network of Biomedical Research Excellence (OK-INBRE) during the Summer of 2010. The motivation for this book was to find material for students who might not be continuing their studies in mathematics but who would need to apply mathematical skills to science courses. Additionally, we designed this book for elementary education majors to satisfy the algebra requirements as recommended by the National Council of Teachers in Mathematics (NCTM). We began the development of this book by having discussions with a variety of science faculty and asking what their expectations were for mathematical skills for students completing a college algebra course. It was enlightening to learn that many of the skills that our science colleagues expected of these students were not skills that were taught in a traditional college algebra course. The topics of estimation, unit conversions, geometry and extrapolation were the skills most often mentioned that were not covered in College Algebra. We have included these topics which makes our book quite different than a traditional algebra text. The other striking difference between our book and a traditional college algebra book is the use of demonstrations and projects. We wanted to motivate the students to study the mathematics by presenting a problem from a field of science that would use the particular mathematics. We have written these demonstrations so that no previous knowledge of the scientific principles involved are required from either the instructor or the student. By introducing the framework first we hope that this will motivate the students to study the mathematics and to address the perennial question “why do I need to learn this?” We include skill based exercises throughout the chapters so that students can check their progress. Instead of the traditional problem sets at the end of the chapter, we have compiled a list of projects that use the skills developed in the chapter. When possible, we have used real data sets and occasionally ask the student to collect the data themselves. The solutions to the projects are intended to be written reports using the mathematics and explaining the procedures and concepts. Typically we assign one or two projects per student. For some of the more involved projects that require data collection we might assign the project to a group. This book is used to teach a one semester course. We cover all of the material in that time. We do use class time for demonstrations, collaborative exercises, and for working with technology. The authors wish to acknowledge and recognize the significant work on the first edition by co-authors Dr. Bryan Clark and Dr. John McArthur. We also ii wish to thank Dr. Diane Dixon for contributing projects for the second edition. iii

Technology Students will need to have access to either computer software or a graphing calculator that does linear, quadratic, and logarithmic or exponential regression. We have used both the statistical software JMP c , the TI-83 Plus graphing c calculator, and Mathematica in teaching this course. In teaching this course with a graphing calculator, it should be noted that starting in Chapter 4 we ask for the adjusted R2 value which the TI-83 Plus does not find. The questions can be modified appropriately but it is important to point out that comparing the R2 value between a linear and a quadratic fit, for example, is not an appropriate way to decide which one of the two curves fits the data better. We have included web addresses for projects that ask students to find additional information before completing the project. However, with the fluidity of the world wide web, these addresses may change and the students might need to find the information at some other site. Supplements There is an instructor’s guide available to instructors who are using this book in teaching their course. It contains all the solutions to the problems and projects when possible. For the projects that rely on students collecting data or other information, we have included a grading guide in the instructor’s manual. Additionally, there is a list of equipment that is needed for the demonstrations and the projects. Appendices We have included three appendices with the book. The first, Appendix A, discusses basic scientific notation. Appendix B is a list of references and web sites for further information. The last appendix contains the answers to the odd problems. iv Contents

1 Estimation 1 1.1 Objectives: ...... 1 1.2 Introduction: ...... 1 1.3 Demonstration: ...... 2 1.4 The Mathematics: ...... 3 1.4.1 NumericalEstimation ...... 3 1.4.2 SamplingTechniques...... 5 1.4.3 UnitConversions...... 7 1.5 Homework Projects: ...... 12

2 Geometry 15 2.1 Objectives: ...... 15 2.2 Motivation: ...... 15 2.3 Introduction to Home Range and Territory: ...... 16 2.4 Demonstration: ...... 18 2.5 The Mathematics: ...... 19 2.5.1 Area...... 19 2.5.2 Perimeter ...... 24 2.5.3 Volume ...... 26 2.5.4 SurfaceArea ...... 29 2.6 Homework Projects: ...... 31

3 Lines 39 3.1 Objectives: ...... 39 3.2 Motivation: ...... 39 3.3 Introduction to Population Size Estimation: ...... 40 3.4 Demonstration: ...... 42 3.5 The Mathematics: ...... 43 3.5.1 Slope ...... 43 3.5.2 TheEquationofaLine ...... 45 3.5.3 FindingSlopesandIntercepts ...... 47 3.5.4 Graphs ...... 48 3.5.5 Functions ...... 50 3.5.6 LinearRegression...... 52

v vi CONTENTS

3.6 Extrapolation and Interpolation: ...... 56 3.7 Homework Projects: ...... 58

4 Quadratics 67 4.1 Objectives: ...... 67 4.2 Motivation: ...... 67 4.3 Introduction to Kinematics: ...... 68 4.4 Demonstration: ...... 69 4.5 The Mathematics: ...... 70 4.5.1 Vertex...... 70 4.5.2 Intercepts ...... 72 4.5.3 Graphs ...... 74 4.5.4 Regression...... 77 4.6 Homework Projects: ...... 83

5 Exponentials 97 5.1 Objectives: ...... 97 5.2 Motivation: ...... 97 5.3 Introduction to Population Growth: ...... 98 5.4 Demonstration: ...... 98 5.5 The Mathematics: ...... 100 5.5.1 IntegerExponents ...... 100 5.5.2 Graphs ...... 105 5.6 Homework Projects: ...... 107

6 Logarithms 115 6.1 Objectives: ...... 115 6.2 Motivation: ...... 115 6.3 Introduction to Sound: ...... 116 6.4 Demonstration: ...... 118 6.5 The Mathematics: ...... 119 6.5.1 Logarithms ...... 119 6.5.2 CommonLogsandNaturalLogs ...... 122 6.5.3 Graphs ...... 123 6.5.4 Equations...... 125 6.5.5 RatesofGrowth ...... 127 6.6 Homework Projects: ...... 130

7 Systems of Equations 139 7.1 Objectives: ...... 139 7.2 Motivation: ...... 139 7.3 Demonstration: ...... 140 7.4 The Mathematics: ...... 141 7.4.1 TwoLinearEquations ...... 141 7.4.2 ThreeLinearEquations ...... 146 7.4.3 NonlinearSystems ...... 149 CONTENTS vii

7.5 Homework Projects: ...... 153

A Scientiﬁc Notation 171

BReferencesandFurtherReadings 173 B.1 Chapter1 ...... 173 B.1.1 BodyMassIndex...... 173 B.1.2 EmpireStateBuilding ...... 173 B.1.3 Molarity...... 173 B.2 Chapter2 ...... 173 B.2.1 Bats ...... 173 B.2.2 CMRMethod...... 174 B.3 Chapter3 ...... 174 B.3.1 Bodyweightversusranges...... 174 B.3.2 TargetHeartRates...... 174 B.4 Chapter4 ...... 174 B.4.1 History ...... 174 B.4.2 Reﬂectivityofwatersurfaces ...... 174 B.4.3 SpeciesDiversity ...... 175 B.4.4 PopulationDynamics ...... 175 B.5 Chapter5 ...... 175 B.5.1 History ...... 175 B.5.2 Population ...... 175 B.5.3 WhoopingCranes ...... 175 B.6 Chapter6 ...... 176 B.6.1 History ...... 176 B.6.2 Titanium-44 ...... 176 B.6.3 ShannonDiversityIndex...... 176 B.7 Chapter7 ...... 176 B.7.1 History ...... 176 B.7.2 Population Growth versus Food Production ...... 176 B.7.3 Censusdata...... 176 B.7.4 OxygenConsumption ...... 177 B.7.5 Bluebirds ...... 177

C Answers to Odd Problems 179 C.1 Chapter1 ...... 179 C.2 Chapter2 ...... 180 C.3 Chapter3 ...... 181 C.4 Chapter4 ...... 185 C.5 Chapter5 ...... 187 C.6 Chapter6 ...... 188 C.7 Chapter7 ...... 190 viii CONTENTS List of Tables

1.1 UnitConversionsfortheU.S.System ...... 7 1.2 PreﬁxesfortheMetricSystem ...... 7 1.3 Conversionsbetweensystems ...... 8

2.1 Capture location data for thirteen-lined ground squirrels by day 18 2.2 UnitConversionsforVolume ...... 26 2.3 Displacement versus compressional force ...... 32 2.4 Rangedataforgroundsquirrels ...... 36

3.1 Number of fox squirrels harvested from the Atoka WMA . . . . 42 3.2 Number of fox squirrels live trapped on the Atoka WMA . . . . 42 3.3 Percentage ofsmokersamong18to25year olds ...... 56 3.4 Winning times in the Olympic women’s 400m freestyle event .. 58 3.5 GPAandMCATscores ...... 59 3.6 Income levels relative to years of school and gender ...... 62 3.7 BodyWeightversusHomeRangeSize ...... 65

4.1 Incomeofhouseholders,2011 ...... 77 4.2 RatioofNIRtoREDversuschlorophyll ...... 84 4.3 Species diversity as related to ﬁre ...... 86 4.4 Boilingtemperaturesofwater ...... 87 4.5 Planetary Orbits and Distance from Sun ...... 88 4.6 AverageClutchSize byLayDateatLatitude30 ...... 89 4.7 AverageClutchSize byLayDateatLatitude45 ...... 90 4.8 AverageClutchSizebyLayDatein1997 ...... 90 4.9 AverageClutchSizebyLayDatein1998 ...... 91

5.1 Estimated size of the human population in the United States .. 99 5.2 Estimated size of the human population in Oklahoma ...... 107 5.3 EvolutionofProcessorSpeed ...... 108 5.4 First U.S. utility patent number in each calendar year ...... 109 5.5 CPUvs. Numberoftransistors ...... 110 5.6 Blood Pressure, Selected Animals ...... 111 5.7 Average Heart Rate, Selected Animals ...... 111

ix LIST OF TABLES 1

5.8 Newton’sLawofCooling ...... 112 5.9 Numbers ofWhoopingCranesat Aransas NWR ...... 113 5.10 Numbers of Bison at Wichita Mountains National Forest . . . . 114

6.1 Estimated size of the human population in the United States . . 128 6.2 EvolutionofProcessorSpeed ...... 130 6.3 CPUvs. Numberoftransistors ...... 131 6.4 First U.S. utility patent number in each calendar year ...... 132 6.5 Newton’sLawofCooling ...... 133 6.6 ActiveUsers atFacebook(inmillions) ...... 134 6.7 Number of small mammals captured in native prairie and im- provedpasturebyspecies ...... 138

7.1 PopulationandFoodProduction ...... 161 7.2 PopulationandFoodProduction ...... 163 7.3 PopulationandFoodProduction ...... 164 7.4 PopulationandFoodProduction ...... 165 7.5 CensusData ...... 166 7.6 PopulationinMexico ...... 167 7.7 PopulationinUnitedStates ...... 167 7.8 AverageClutchSizebyLayDate ...... 168 7.9 AverageClutchSizebyLayDate ...... 169 7.10 Time (minutes) on treadmill and power exerted (watts) . . . . . 170 7.11 Power (watts) exerted vs. oxygen consumed (L/min) ...... 170 Chapter 1

Estimation

1.1 Objectives:

After completing this chapter, you should be able to

Estimate answers in problems where exact solutions are not needed • Estimate the size of things that are too large or diﬃcult to measure • Estimate the number of objects when it is not possible or practical to • count them Convert basic units of measurement • 1.2 Introduction:

Using mathematics in everyday life requires that one possess good estimation skills. Questions that might come up, for example, if you are going to the grocery store:

Which is the better buy, a 12-pack of 12 oz. cans of soda for $3.99 or a 2-liter bottle for $1.28? or to a restaurant:

Is $25.00 enough if the steak is $14.95, the drink is $2.50, the tax rate is 8% and you want to leave at least a $3.00 tip?

1 2 CHAPTER 1. ESTIMATION or working at home:

If one can of paint covers 200 square feet, about how many cans 1 would you need to put 2 coats on a wall that is 8 2 feet tall and 19 feet long? or on a trip:

If it is 212 miles to your destination and you have half a tank of gas, do you need to stop to get more gas? or having a party:

About how many pizzas does it take to feed 10 people?

Notice that some of the above examples include the word “about.” It is not always imperative that we get an exact answer. Often an estimation is all we need. We don’t care that it takes 2.56 cartons of tile to cover the kitchen ﬂoor because we can’t buy a fraction of a carton anyway. We just want to know if we should buy 2 or 3 cartons. If a 15% tip is $1.89, we will probably just leave $2.00. However, we do need to know that our answer is approximately right. We don’t want to buy 5 cans of paint when only 3 are needed or buy cans of soda when the cost is twice as much per ounce as a 2-liter bottle. There are also many instances in science where estimation must be used because exact answers are not possible. Suppose, for example, that we want to know the population size of a particular species of animal; or the volume of a lake or river; or the distance between two planets. In each of these cases an exact answer cannot be found and an estimation will usually suﬃce.

1.3 Demonstration:

Exploration Question #1: We would like to know how many trees there are on your campus. We probably can’t count all of the trees on campus (at least not in a reasonable amount of time). Instead we will have to settle for estimating the number. Discuss in your group what strategy you could use to estimate the number. Discuss what assumptions you are making to use that strategy. Discuss what preliminary information you would need to know. Be prepared to share your answers with the rest of the class and to possibly put your strategy into action.

Exploration Question #2: Each person should write up in complete sentences the strategy their group decided on. Try to do your write up without discussing it with anyone in your group. Make sure to write it so that someone else who did not participate in your discussion could follow your steps and could use your strategy to 1.4. THE MATHEMATICS: 3

estimate the number of trees on campus. Explain your assumptions, what preliminary information you would need, and how you could go about obtaining that information.

Exploration Question #3: After everyone has ﬁnished their report, exchange your report with someone in the class who was not in your group. Using your partner’s report, perform the strategy that is described. If you do not understand any part of the strategy, mark on the paper what you do not understand or what additional information you need. Try to perform the strategy exactly as it is written up.

1.4 The Mathematics: 1.4.1 Numerical Estimation Numerical estimation involves rounding numbers that appear in a particular problem. The usual rule for rounding is that if the next digit is 5 through 9, you round up. For digits 1 through 4 you round down. You also need to know the level of accuracy needed. For example, 12,582 could be rounded to the nearest tens (12,580), the nearest hundreds (12,600), the nearest thousands (13,000) or the nearest ten-thousands (10,000). The context and the calculations usually determine which is the best rounding place. Additionally, the rounding rules may be changed in situations where it would have a big negative impact if your rounding is slightly oﬀ. For example, if you round all of your checks to the nearest dollar in your checkbook, you might decide to round up at 4, since if your estimate is too low and you bounce a check, there will be a ﬁnancial penalty. Exercises : Round the following numbers to the nearest tenths, ones, tens, and hundreds.

1. 1,305.69

2. 34 3. 689,560.4893 4. 0.7297 5. 164.2 6. 78,520.739 7. 3.14159 4 CHAPTER 1. ESTIMATION

Let’s work through some of the situations that were mentioned in the introduction. Example 1: “Which is a better buy, a 12-pack of 12 ounce cans of soda for $3.99 or a 2-liter bottle for $1.28?” Sometimes you might not have enough information given to you in an estimation problem and it will be necessary to find the information elsewhere. In this example, to determine which is the better buy it is necessary to know how much soda you get in a 12-pack of 12 ounce cans versus a 2-liter bottle. In the volume section in the next chapter we will discuss how to convert ounces to liters and vice versa. Example 2: “Is $25.00 enough money if the steak is $14.95, the drink is $2.50, the tax rate is 8% and you want to leave at least a three dollar tip?” It is probably best to round to the nearest dollar for the steak. We know that we will spend about $15.00 for the steak, $2.50 for the drink, and $3.00 for the tip, which is a total of $20.50. So, we need to know if $4.50 is enough for the tax. Since the tax rate is 8%, it will be easiest for us to calculate if we round this to 10%. Tax is not computed on tips so we need to calculate 10% of $17.50. To find a percent of something we multiply by the equivalent decimal representation. So, to find 10% of $17.50, we need to calculate 0.10 17.50 which we can do in our head by moving the decimal place over one place× to the left. Thus the tax will be about $1.75 on $17.50, so we should have enough money. Again, this is one situation where you probably should round up and overestimate rather than underestimate. If you don’t have enough money, you don’t want to end up doing dishes! Example 3: “If one can of paint covers 200 square feet, about how many cans 1 would you need to put 2 coats of paint on a wall that is 8 2 feet tall and 19 feet long?” In this example, we are going to need to estimate the area of the wall. In the next chapter we will discuss area in more detail, but for a rectangular wall the 1 area is given by length times width. It might be easier to round the 8 2 feet to 9 or 10 and the 19 feet to 20. Whether or not you should round probably depends on if you have a calculator handy and how close you want your calculation to be to the actual answer. If we do round to 10 feet and 20 feet, the area will be 10 feet 20 feet = 200 square feet. Since one can of paint covers 200 square feet we× know we will need 2 cans, one for each coat. 1.4. THE MATHEMATICS: 5

Example 4: “If it is 212 miles to your destination and you have half a tank of gas, do you need to stop to get more gas?” Here is another example where not enough information is given to answer the question. Information that is missing, and that would change from one person’s car to the next person’s car, is “how many gallons of gas does the tank hold?” and “how many miles per gallon does your car get in gas mileage?” Until we know the answers to these questions, we won’t be able to answer this question. Example 5: “About how many pizzas does it take to feed 10 people?” Again, although this sounds like a simple question, we need more information. For example, we need to know how big the pizzas are. We need to know if the crust is thick or thin, since most people eat fewer slices of a thick crust pizza than of a thin crust pizza. It also might make a diﬀerence who the 10 people are. If we are talking about 10 football players after a heavy work out, we might need to buy more pizzas than if we are talking about 10 math club members (but then again, maybe not). Exercises : For the following questions, decide if you are given enough information to answer the question. If you are, perform the estimation. If you are not, list all of the additional information that you would need to answer the question.

8. About how much would a 15% tip be for a meal that costs $23.52? 9. If you are painting the ﬂoor of a circular room with a radius of 9 feet, how many cans of paint would you need? 10. Is $6.00 enough to buy milk, eggs, and bread? 11. If it is 332 miles to your destination and you need to be there in 6 hours, about how fast do you need to drive? 12. About how long would it take to drive from Chicago to Los Angeles if you average 70 miles per hour?

1.4.2 Sampling Techniques Another type of estimation problem involves taking a small sample and then extrapolating to a larger region or population. For example, if there is a contest to see who can come closest to guessing the correct number of candies in a jar, you won’t be able to actually count the candies. One way to estimate the number of candies is to pick some portion of the jar that you can see and determine the number of candies in that portion. Using this data, it is then possible to extrapolate to the entire jar. If you were just counting green candies, for example, you could use the number of green candies in your sample to estimate the number of green candies in the jar. One shortcoming of this approach is 6 CHAPTER 1. ESTIMATION that your sample region may or may not be representative of the whole area of interest. One way to account for this shortcoming is to make sure you takea random sample. A random sample is one in which each member of the whole has an equal chance of being included in the sample. If your sample is not random, then it may be biased one way or another. For example, if most of the green candies are at the bottom of the jar and we only sample from the top, we will greatly underestimate the number of green candies in the jar. Even if a random sample of people is chosen, it may be that not everyone in the sample will respond. In this case the people who respond to a survey may not be representative of all people. Many times people who feel strongly about an issue (either negatively or positively) are more likely to respond to a survey on that issue that those that are neutral. In this case the sample has what is called a volunteer bias. In most cases, the randomness of the sample is actually more important than the size of the sample. For example, the Nielsen ratings for television shows are based on a sample of only 5,000 families. The information from these families is then used to estimate the entire nation’s viewing habits. Nielsen Media, the company that determines the Nielsen ratings, employs a team of statisticians whose job is to make sure that the 5,000 families are selected randomly. Exercises : For each of the situations described below tell whether you think the sample is biased or not and justify your answer.

13. You want to know the average age of students at your university. You use as your sample the students that are in your algebra class. 14. A popular magazine company wants to know what percentage of couples say they are happily married. Their sample consists of readers that fill out a survey in the magazine and then mail it back to the company. 15. A teacher needs to choose 4 students out of 20 to go on a field trip. She puts 16 red marbles and 4 blue marbles in a hat and mixes them up. Each student draws a marble out without looking. The ones that draw out blue marbles get to go on the trip. 16. A bookstore would like to know how satisfied their customers are with the purchasing experience. To find out, they ask their customers who have made an online purchase to fill out a survey. 17. A politician would like to know how her constituents feel about a particular issue. She sends out a survey through the mail to find out. 1.4. THE MATHEMATICS: 7

1.4.3 Unit Conversions Many times, whether it is in this book, in your science classes, or in real life, it is necessary to convert between basic units of measurements. There are two basic systems of measurements - the U.S. system and the metric system. In addition to converting between units in one system, it will also be necessary to convert between the two systems. Table 1.1 gives some basic conversions in the U.S. system.

1 foot 12 inches 1 yard 3 feet 1 mile 5280 feet

Table 1.1: Unit Conversions for the U.S. System

In the metric system, the basic unit of measurement for length is the meter and the basic unit of measurement for mass is the gram. To understand the metric system, it is necessary to know the meaning of the prefixes. Table 1.2 gives some of the common prefixes and their meanings. It is useful to write these numbers in scientific notation. If you are not familiar with scientific notation, please refer to Appendix A.

giga billion 1,000,000,000 1 109 × mega million 1,000,000 1 106 × kilo thousand 1,000 1 103 × hecto hundred 100 1 102 × deka (deca) ten 10 1 101 1 × 1 deci one-tenth ( 10 ) 0.1 1 10− 1 × 2 centi one-hundredth ( 100 ) 0.01 1 10− 1 × 3 milli one-thousandth ( 1,000) 0.001 1 10− 1 × 6 micro one-millionth ( 1,000,000) 0.000001 1 10− 1 × 9 nano one-billionth ( ) 0.000000001 1 10− 1,000,000,000 ×

Table 1.2: Preﬁxes for the Metric System

1 So, for example, a kilogram is 1,000 grams or a centimeter is 100 of a meter. 8 CHAPTER 1. ESTIMATION

It is helpful to have some basic feel for comparisons between the U.S. system and the metric system. A meter stick is just a little bit longer than a yard stick. Since a yard stick is 3 feet long, and each foot has 12 inches, a yard stick is 36 inches long. A meter is approximately 39 inches. The mass of a nickel is about 5 grams. Table 1.3 gives some basic conversions between the two systems.

1 meter 39.37 inches ≈ 1 kilogram 2.2 lbs ≈

Table 1.3: Conversions between systems

To convert between diﬀerent units, whether it is between units in the U.S. system, or between units in the metric system, or converting between the two systems, it is probably easiest to think of unit conversions as ratios. For example, we can write the fact that 1 yard is 3 feet as a ratio in two diﬀerent ways: 1 yard 3 feet or . 3 feet 1 yard We will want our units to cancel and this will determine which way we will write this ratio. If we want to determine how many yards are in a mile using this ratio technique we would need the following two ratios (from Table 1.1):

1 mile 3 feet and 5, 280 feet 1 yard We just need to make sure that the units for feet will cancel since for our ﬁnal answer we are interested in miles and yards. Multiplying these two ratios together we get 1 mile 3 feet 1 mile = 5, 280 feet · 1 yard 1, 760 yards since 5,280 divided by 3 is 1,760. So this means that there are 1,760 yards in a mile. If we know how many yards are in a mile, then we can ﬁnd out how many yards are in a set number of miles by multiplying. For example, if there are 1,760 yards in 1 mile, then there would be 5 1, 760= 8, 800 yards in 5 miles. Converting between the two systems is a∗ similar process except that your answers will be approximations instead of exact answers. For example if we wanted to convert kilometers to feet, we would need the following ratios: 1 kilometer 1 meter 12 inches . 1, 000 meters · 39.37 inches · 1 foot ≈ 1.4. THE MATHEMATICS: 9

Since the second ratio is given by an approximation, our answer will also be an approximation. Notice that the ratios were arranged so that meters and inches cancel leaving us with kilometers and feet which is what we want. If we do the arithmetic, we have 1 kilometer 3, 281 feet ≈ or that there are approximately 3,281 feet in a kilometer. If we wanted to know approximately how many kilometers are in a foot, we would need to divide 1 by 3,281 to get

1/3, 281 kilometer ≈ . 1 foot This works out to be 0.0003047. The answer is less than 1 in this example since a kilometer is much≈ longer than a foot. If we want to convert ﬁve kilometers to feet, we can do a similar process to the above, except we start with 5 kilometers and arrange our ratios so that the appropriate units cancel. 1000 meter 39.37 inches 1 foot 5 kilometers ≈ . · 1 kilometers · 1 meter · 12 inches Multiplying 5 times 1000 times 39.37 and then dividing by 12 gives us that there are approximately 16,404 feet in ﬁve kilometers. Since we know from above that there are approximately 3,281 feet in a kilometer, we can check our answer by multiplying 3,281 feet by 5. Since we had rounded, our two answers are close but not exactly the same. 10 CHAPTER 1. ESTIMATION

Exercises :

18. How many meters are in a kilometer?

19. How many kilometers are in a meter?

20. Convert 5 grams to centigrams.

21. Convert 5 centigrams to grams.

22. How many millimeters are in a hectometer?

23. How many megaliters are in a gigaliter?

24. How many kilometers are in a mile?

25. How many inches are in 3 miles?

26. How many miles are in 3 yards?

27. How many centimeters are in an inch?

28. How many seconds are in an hour?

29. How many seconds are in a year?

30. Convert 37 years to seconds.

31. Convert 3 miles to yards.

32. Convert 3 kilometers to meters.

We can also use the ratio method to convert units that have more than one component. For example, we might be concerned about how fast we are driving. In the United Stated this is measured in miles per hour. The word “per” in this instance means that written as a fraction, the hours belong in the denominator. We might be driving 60 miles per hour, which is written as

60 miles . 1 hour If you are driving in a British Commonwealth country such as England or Canada, however, speed (as well as the speed limit) is measured in kilometers per hour. To ﬁgure out how fast 60 miles per hour is in kilometers per hours we have: 60 miles 1 kilometer 5, 280 feet 96 kilometers . 1 hour · 3, 281 feet · 1 mile ≈ 1 hour ≈ The next time you are driving 60 miles per hour, check your speedometer to see what the kilometers per hour reading is. 1.4. THE MATHEMATICS: 11

We will also need to be able to convert both the length measurement and the time measurement. If in the above example we needed to know how fast we were driving in kilometers per minute, we would add the following step: 96 kilometers 1 hour 1.6 kilometers . 1 hour · 60 minutes ≈ 1 minute

Exercises :

33. Convert 1 mile per hour to meters per second. 34. Convert 1 foot per second to meters per second. 35. Convert 1 kilowatt hour to watt hours (Hint: This is not a “per” problem.) 36. Convert 1 kilowatt hour to watt seconds (Hint: This is not a “per” problem.)

37. If a car is traveling at a rate of 70 mph, how many feet will it go in 10 seconds? 38. Health oﬃcials and nutritionists use the Body Mass Index (BMI) as a measurement to assess a person’s degree of obesity. Since it is only based on your height and weight, it does not take into account gender, age, or the body composition percent of fat. A healthy BMI is considered to be between 19 and 25. The formula for the BMI is mass in kilograms . (height in meters)2

Find your weight in pounds and your height in feet and inches and use this (with the proper conversion factors) to calculate your BMI. 12 CHAPTER 1. ESTIMATION 1.5 Homework Projects:

For each assigned project, write your answer in complete sentences. Projects will be graded based on thoroughness and neatness.

1. In this project we are going to compare your estimated caloric input with your estimated caloric needs. To begin, estimate the number of calories that you eat in a day by turning in a list of all the food you ate for one day and the calorie count for each serving. Indicate the source(s) that you used to determine your calorie counts. How close do you think your estimation is to the actual number of calories you eat in a typical day? To ﬁgure out your resting energy expenditure (REE), you need to calculate your weight in kilograms and your height in centimeters. Find (and write down) your weight in pounds and then convert to kilograms. Find (and write down) your height in feet and inches and then convert this to centimeters. Use the following formulas to calculate your REE:

MaleREE = 10 weight(kg)+6.25 height(cm) 5 age +5 × × − ×

F emaleREE = 10 weight(kg)+6.25 height(cm) 5 age 161 × × − × −

According to some health sources, REE 1.5 is an appropriate starting point for caloric estimation for weight maintenance.× To ﬁnd the number of calories you should be consuming to lose weight you should multiply your REE by a number from 1.1 to 1.3. If you want to lose weight more quickly, use REE 1.1 and if you want to lose weight more slowly, use REE 1.3. Using× your estimation for the number of calories you eat in a typical× day, do these formulas for weight maintenance and weight loss indicate that you are maintaining your weight? Explain.

2. Estimate the number of utility poles in the county that you live in by taking a sample of roads within the county. For each portion of the road sampled, count the number of utility poles along the road. Make sure to include your data, an explanation of how you determined your sample, and how you determined the number of miles of road in the county.

3. You have been put in charge of planning a picnic for the faculty and their families for your university. You are going to serve hamburgers, hot dogs, chips, pop, and cookies. Estimate how much you need of each of the above items for the picnic. Explain your assumptions and your method for estimation. 1.5. HOMEWORK PROJECTS: 13

4. Describe a strategy for estimating the number of road kill mammals in the county that you live in on any given day. What are the assumptions behind your strategy? What are its advantages and disadvantages? How could it be improved? Using your strategy, estimate the number of animals killed in a typical day.

5. Describe a strategy for estimating the thickness of a sheet of paper. De- scribe the type of paper that you are using and estimate the thickness. Use this estimate to approximate the number of sheets of paper you would need to stack to reach the height of the Empire State Building.

6. Estimate the percentage of high school baseball players that eventually end up playing in the major league in the United States. Explain your assumptions and your method for estimating this number.

7. Estimate the number of words in this book. Explain your assumptions and your method for calculating this number.

c 8. Estimate the number of chocolate chips in a bag of Chips Ahoy cookies. Explain your assumptions and your method for calculating this number.

9. Describe a strategy to estimate the total number of commercials that are shown on one channel in one day. (Please indicate which channel you picked.) Explain the assumptions of your strategy. Using your strategy, estimate the total number of commercials that are shown on your selected channel over a 24 hour period.

10. Describe a strategy to estimate the number of pounds of newspaper that are delivered in your city in one week. Explain the assumptions of your strategy. Using your strategy, estimate the number of pounds of newspaper that are delivered in your city in one week.

11. Describe a strategy to estimate the number of books on the shelves in the library. Explain the assumptions of your strategy. Using your strategy, estimate the number of books on the shelves in the library. If it is possible, compare your estimate with the actual count of library books on the shelves.

12. Describe a strategy to estimate the number of parking spaces on campus. Explain the assumptions of your strategy. Using your strategy, estimate the number of parking spaces on campus. If it is possible, compare your estimate with the actual number of parking spaces on campus. 14 CHAPTER 1. ESTIMATION

13. (Note: This project was originally developed in conjunction with Dr. Di- ane Dixon. See Appendix B.) A mole is a unit in chemistry which is related to the molecular weight of a chemical. For example, the molecular weight of sodium chloride (NaCl) is 58.5; therefore a mole of NaCl would be 58.5 grams. Molarity is a measure of a solution’s concentration using moles. A one molar solution is one mole of a chemical in a liter of solvent (which is generally water). For example, if you wanted a 2 molar solution (2M) then it would be 2 moles of the chemical in a liter of water. If you only wanted to make 500 milliliters (mL) of the solution, then you would add 1 mole of the chemical to 500 mL (0.5L) of water (1 mole/0.5L = 2M).

(a) Explain how you would make a 100 mL solution of 5M NaCl. (b) Explain how you would make a 250 mL solution of 2M Tris. (Tris is a chemical used for buﬀering in biology. It has a molecular weight of 121.) Many times in science a stock solution of a high concentration is made and dilutions are made of that to make lower concentrations. For this example, we could use the 5M NaCl solution and the 2M Tris solution to make a 500 mL (0.5L) 0.025M Tris, 0.15 M NaCl solution. (c) How much of the 2M Tris solution would be needed to make 500mL of a 0.025M Tris solution? (d) How much of the 5M NaCl solution would be needed to make 500mL of a 0.15M NaCl solution? (e) How would you make the 500 mL 0.025 Tris, 0.15 M NaCl solution? (Make sure it is a 500 mL solution.) Chapter 2

Geometry

2.1 Objectives:

After completing this chapter, you should be able to

Calculate the area and perimeter for common two dimensional objects • Calculate the volume and surface area for common three dimensional ob- • jects

Estimate area and perimeter for irregularly shaped two-dimensional ob- • jects

Estimate volume and surface area for irregularly shaped three-dimen- • sional objects

2.2 Motivation:

Geometry has been studied and used for literally thousands of years. In ancient times it was used for navigation, surveying, architecture, and cartography (map- making) in places as diverse as Greece, India, and Iran as well as many other cultures. It is still used for these purposes today along with many other more modern applications. In science, geometry is used in current research in areas such as: virtual reality, molecular modeling, protein structure, cell growth simulation, asteroid surface mapping, medical imaging, and signal processing (e.g., CAT-scans). In manufacturing one might be interested in how to cut out a collection of shapes with as little waste of material as possible, or there may be an interest in the number of ways a piece of sheet metal can be folded. In business, one may need to determine the area covered by a cellular phone transmitter or the best way to install a sprinkler system where each sprinkler head covers a circular area. These questions are all answered with geometry. Geometry can also play a role

15 16 CHAPTER 2. GEOMETRY in everyday life, whether one is making a quilt, tiling a bathroom, or painting a picture. As you can see, geometry has a variety of applications and more are being discovered nearly every day. In this chapter we will explore the use of geometry in the biological and physical sciences. In the next section we will discuss how the concepts of area and perimeter are applied to wildlife management.

2.3 Introduction to Home Range and Territory:

Most people are familiar with the concepts of home range because they have observed the same animal in the same general area on many occasions. Biologists define home range as the area an animal uses during its lifetime to find food, mates, and care for young. Home ranges, at least in some instances, also may have an important third dimension because of vertical movement (e.g., use of trees and shrubs by birds or use of underground burrows by gophers); however, this concept of home range is beyond the scope of this book. In some instances, the entire home range is not used each day and the animal’s activities occur in specific locations called core areas. If an animal maintains an exclusive area and exhibits aggressive behavior when another member of the same species enters its home range, or a portion of the home range, the defended area is called a territory. For example, most people have heard either their cat, or a neighbor’s cat, fighting with a stray cat to defend its territory. Therefore, home range and territory are related but differ because an animal may not defend the entire area it uses. Animals must find food to survive and several researchers have documented the increase in home range size with increased body weight. For example, least shrews which weigh less than 10 grams have a home range size of less than 0.6 hectares. In contrast, white-tailed deer, which may weigh over 90 kilograms, have a home range size of greater than 800 hectares. Why do we need to know about home range size and its relationship to body weight? With the increasing human population, many animals are found primarily on areas managed by private, state, or federal agencies. To determine what area must be protected, information about the space-use requirements of focal species must be known. For example, there has been considerable interest in reintroducing the gray wolf into the lower 48 states. Since the home range of a pack of gray wolves has been estimated to be between 125 km2 and 2,541 km2, the Greater Yellowstone ecosystem is one of the few places which is protected enough and large enough for reintroduction of this species. An animal’s movements can be monitored using various methods. One such method is called capture-mark-recapture (CMR), which would involve setting live capture traps on a grid. An animal caught for the first time would be uniquely marked, its capture location recorded, and then released. Every time the marked animal is recaptured at some later date the location would be recorded and this information would be used to estimate the size of the home range. Mammalogists have used live-trap grids for many years to study various 2.3. INTRODUCTION TO HOME RANGE AND TERRITORY: 17 species of small mammals and therefore, procedures to estimate home range are well established. Two standard methods for estimating the size of a home range include:

1. Minimum area - Capture locations of an animal are connected in a way such that every capture location is connected by a line segment to exactly two other capture locations and such that no line segments cross. Start- ing with the top left hand point, proceed in a counterclockwise manner, connecting points so that the enclosed region has the minimal area (see Figure 1). 2. Range length - Distance between the two most widely separated captures (see Figure 1).

Figure 1: Capture locations of a thirteen-lined ground squirrel (Squirrel 1 from Table 1). 18 CHAPTER 2. GEOMETRY 2.4 Demonstration:

Use the capture-mark-recapture data in Table 2.1 for thirteen-lined ground squirrels to complete the exercises listed below. Data for ground squirrels were collected on an 18 18 station live-trap grid. The traps were set on a grid with one axis aligned North× to South and the other aligned East to West. The spacing between traps in each column and each row was 15 meters. The columns are denoted by letters A through R and the rows 1 through 18.

# Day1 Day2 Day3 Day4 Day5 Day6 Day7 Day8 Day9 Day10 1 A5 A8 B10 C5 B7 D4 A9 E5 D9 E7 2 G12 G14 L17 M16 J13 H17 I12 P14 Q17 G16 3 M2 J5 R2 N9 P8 R5 L7 N9 O1 L4 4 H8 G7 E9 E11 F13 G14 I14 I11 K10 J7 5 C13 E13 F13 I15 G18 D18 C17 A16 B14 H17 6 M1 M3 K4 K7 P1 P2 Q3 O4 N5 N7 7 E1 F1 C2 B3 C4 F3 F3 I3 F5 D5 8 F7 F9 F9 G11 E11 H7 I11 I8 I8 J10 9 K15 K15 J13 H12 F12 E13 E15 I16 H16 F17 10 R10 R12 P10 O9 Q14 P15 N15 L12 M14 M11

Table 2.1: Capture location data for thirteen-lined ground squirrels by day

1. For each animal, diagram the home range using the minimum area method. Label columns (A-R) along the top of your paper with each letter placed on a separate line (not the space between two lines) and label rows (1-18) along the left side of your paper with each number placed on a separate line. The letter-number designations for captures of thirteen-lined ground squirrels listed above should be plotted at the intersection of the column and the row.

2. Calculate the range length for each animal. Use the formula a2 + b2 = c2, where a and b are the sides of a right triangle and c equals the hypotenuse of that right triangle.

3. Estimate the area of each animal’s home range. Remember, each trap is separated from the next closest trap by 15 m. Therefore, the sides of each square on your graph paper represent 15 m in the real world and the area for one square is 225 m2 (length width; 15m 15m). × × 2.5. THE MATHEMATICS: 19 2.5 The Mathematics:

2.5.1 Area

Area is a measurement of the size of an enclosed region. Because it has two dimensions, the units for areas are usually square units such as square feet or square kilometers, one for each dimension. We write this mathematically as ft2 or km2. The two commonly used units for areas that are not written as square 1 units are acre and hectare. An acre is 640 of a square mile and a hectare is 10,000 m2. Here is a list of some common two-dimensional shapes and the formulas for their areas, A.

Rectangle:

A = a b · Square: (A square is just a rectangle where the length and width are equal)

A = x x = x2 · General Triangle:

1 A = 2 bh 20 CHAPTER 2. GEOMETRY

Right Triangle:

1 A = 2 ab (The height of a triangle is the shortest distance from the base to the opposite vertex. For a right triangle the height is also one of the sides.) Circle:

A = πr2 (π is an irrational number. It is approximately 3.14159... Depending on the context, you will either leave π in your answer or round off accordingly.) You can find the area of most other shapes by combining these basic formulas. It might also be necessary to use the Pythagorean formula for right triangles that we used to find the range length in the demonstration. If you know two sides of a right triangle, you can use the formula a2 + b2 = c2, where a and b are the shorter sides of the triangle and c is the side opposite the right angle, to find the length of the third side. Exercises : For problems 1-8, find the area of each the following.

1. A rectangle with a width of 3 inches and a height of 2 inches. 2. A circle with a radius of 2 feet. 3. A triangle with a base of 4 meters and a height of 6 meters. 4. A standard sheet of paper (8.5 inches by 11 inches), in square inches. 5. A right triangle with a base of 3 feet and a hypotenuse (the longest side) of 5 feet. (Hint: Use the Pythagorean formula.) 2.5. THE MATHEMATICS: 21

6. A rectangle with a semicircular top as shown with the following dimensions:

7. A parallelogram as shown with the following dimensions:

8. A trapezoid as shown with the following dimensions:

9. How many square feet are in a square mile?

10. How many square miles are in an acre?

11. How many acres are in a square mile?

12. How many acres are in a hectare?

13. Convert 4 hectares to square miles.

14. Convert 2.5 square miles to square yards.

15. Must an acre be in the shape of a square? Explain. 22 CHAPTER 2. GEOMETRY

As we saw in Section 2.3, you might need to ﬁnd or estimate the area of a region that is not composed of circles, rectangles, and triangles. Another place where this occurs is ﬁnding the area under a graph. This situation is common in the physical sciences. For example, suppose we have a graph that for the horizontal axis has the time (in hours) traveled by a bicycle and the velocity (in miles per hour) of the bicycle for the vertical axis (see Figure 2).

18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0.5 1 1.5 2 2.5 3 Figure 2

The distance traveled is the area between the horizontal axis and the curve. Calculus is used to ﬁnd the exact area, but even without calculus, we can approximate the area using rectangles. For this graph we could draw the following 3 rectangles and compute their area to approximate the distance (see Figure 3).

18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0.5 1 1.5 2 2.5 3 Figure 3

If we wanted a better approximation, we could use smaller rectangles. 2.5. THE MATHEMATICS: 23

Exercises :

16. Approximate the area under the curve using 3 rectangles (see Figure 3). Do you think your answer is larger or smaller than the actual area? Ex- plain. 17. Approximate the area under the curve using 6 rectangles (see Figure 4). Do you think your answer is larger or smaller than the actual area? Ex- plain. 18. What other methods could you use (besides using more rectangles) to make your approximation more accurate? Explain.

18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0.5 1 1.5 2 2.5 3 Figure 4 24 CHAPTER 2. GEOMETRY

2.5.2 Perimeter The perimeter of a region is the length of its boundary. Imagine yourself stand- ing at one corner of a rectangle. If you walked along the outside of the rectangle and measured how far it was until you returned to your starting corner, that measurement would be the perimeter of the rectangle. Since you are just measuring length, perimeter is a one-dimensional measurement, so the units would be in feet or meters, for example. Here’s a list of the perimeter (P ) formulas for a rectangle, square, triangle, and circle. Rectangle:

P = a + b + a + b =2a +2b Square:

P = x + x + x + x =4x

Triangle:

P = a + b + c 2.5. THE MATHEMATICS: 25

Circle:

C =2πr (The Latin-based word circumference is usually used instead of perimeter for circles. It is still the measurement around the outside of the circle.) Exercises : Find the perimeter or circumference for each of the following ﬁgures.

19. A rectangle with a width of 3 inches and a height of 2 inches. 20. A circle with a radius of 2 feet. 21. A triangle with a base of 4 meters and sides of 6 meters and 5 meters. 22. A rectangle with a width of 12 inches and a diagonal of 13 inches. (Hint: Use the Pythagorean formula.) 23. A rectangle with a semicircular top as shown with the following dimensions:

24. A parallelogram as shown with the following dimensions: 26 CHAPTER 2. GEOMETRY

25. A trapezoid as shown with the following dimensions:

26. Karla wants to wrap the following package and needs 6 additional centimeters to tie a bow. How long a ribbon should she cut for the package?

2.5.3 Volume The measurement of the quantity of material or space in an enclosed three- dimensional region is called volume. Because it has three dimensions, the units for volume are usually cubic units such as cubic feet or cubic meters. We write these mathematically as ft3 or m3. Measuring volume is complicated, however, by the use of units such as gallon, liter, quart, and pint. It is also necessary to know, for instance, if you are measuring a dry quart (as in a quart of strawberries) or a liquid quart (as in a quart of milk). Table 2.2 lists some conversions between the various units for measuring volume.

1 liquid gallon 4 liquid quarts or 231 cubic inches ≈ 1 dry gallon 227.420 cubic inches ≈ 1 liquid quart 2 liquid pints or .946 liquid liter ≈ 1 liquid pint 16 ﬂuid ounces or 28.875 cubic inches ≈ 1 ﬂuid ounce 1.804 cubic inches ≈ 1 liquid liter 1.056 liquid quarts ≈

Table 2.2: Unit Conversions for Volume 2.5. THE MATHEMATICS: 27

Here’s a list of some common three-dimensional shapes and the formulas for their volumes, V.

Rectangular Solid:

V = a b c · · Cube: A cube is a rectangular solid where the length, height and width are equal.

V = s s s = s3. · · Sphere:

4 3 V = 3 πr 28 CHAPTER 2. GEOMETRY

Right Circular Cylinder:

V = πr2h Note that for the rectangular solid, cube, and right circular cylinder that the volume is the area of the base times the height.

Exercises : In problems 27-31, ﬁnd the volume of the described object.

27. A rectangular solid which has a length of 3 feet, a width of 4 feet, and a height of 2 feet.

28. A cube which has a length of 2 meters.

29. A right circular cylinder which has a radius of 2 cm and a height of 5 cm.

30. A sphere which has a radius of 4 inches.

31. A cereal box that is 12 inches tall and has a 3 inch by 10 inch rectangular base.

32. How many cubic inches are in a cubic foot?

33. Convert 32 ﬂuid ounces to (liquid) liters.

34. Convert cubic kilometers to cubic meters.

35. How many cubic inches are in a (liquid) quart?

36. How many ﬂuid ounces are in a (liquid) gallon?

37. Which is the better buy: A 12 pack of 12 (ﬂuid) ounce cans for $3.99 or a 2 liter bottle for $1.28? Find the cost per ounce for the 12 pack and the cost per ounce for the 2 liter bottle.

38. Water in aquifers is usually reported in units of acre-feet. How many gallons of water is in 1 acre-foot? 2.5. THE MATHEMATICS: 29

2.5.4 Surface Area Surface area, S, is the two-dimensional analogue to perimeter. It is the measurement of the area on the surface of a three-dimensional shape. One way to think of surface area is as the amount of material needed to cover the three- dimensional shape. Since it is an area, the units of measurement are the same as for area, for example square feet or square kilometers. For surface areas, it is useful to understand how the formulas are derived so that you can apply them to diﬀerent situations. Rectangular Solid: To make a rectangular box, with both a top and a bottom, you need 6 rectangles (since it has 4 sides). The area of 2 of the sides is a b; the area of the other two sides is a c. The area of the top and bottom are b · c, so if we add all 6 of the sides together· we have S =2ab +2ac +2bc. ·

Cube: Since all the sides have the same length, the area for each of the sides is s2. For a cube that has a top and a bottom, S =6s2.

Sphere: Here it is probably easier to just memorize the formula for surface area. It is S =4πr2. 30 CHAPTER 2. GEOMETRY

Right Circular Cylinder: If a right circular cylinder has a top and a bottom there are three pieces whose area you need to calculate; the top, bottom, and side. The top and bottom are circles so each of their areas is πr2. The side is a little more complicated. Think of cutting up the side of the can and then unrolling it. You should get a rectangle with its height the height of the cylinder. What about the width? Since you unrolled the cylinder, you are using the circumference of the circle for the width, or 2πr. The area of this rectangle is then 2πrh. Combining these three pieces, the surface area of a cylinder is S =2πrh +2πr2.

Exercises :

39. Find the total surface area of the walls of a room that is 12 feet long, 10 feet wide, and 8 feet high. There is one door which is 7 feet by 2 feet and 2 rectangular windows that are 5 feet by 4 feet. 40. Find the surface area of a rectangular solid which has a length and width of 3 feet by 4 feet and a height of 2 feet if a. the rectangular box has both a bottom and a top. b. the rectangular box has a bottom but no top. 41. Find the surface area of a right circular cylinder with a radius of 2 cm and a height of 5 cm if a. the right circular cylinder has both a bottom and a top. b. the right circular cylinder has a bottom but no top. 42. Find the surface area of a sphere that has a radius of 4 inches. 43. Give the dimensions of two diﬀerent rectangular solids that have the same volume. Do they also have the same surface area? 2.6. HOMEWORK PROJECTS: 31 2.6 Homework Projects:

For each assigned project, write your answer in complete sentences. Pro-jects will be graded based on thoroughness and neatness as well as on correctness. Make sure to show any calculations that you used in your project.

1. Pick three of the squirrels from the capture and location data from Ta- ble 2.1 of the Demonstration section. Diagram the home range, disregard- ing any “interior” points. So, when you connect the dots, you may have dots inside your region that are not connected to any other dots. For these polygons, calculate the animal’s home range and range length. For the range length, indicate which two points you are using. Diagram the home range for the same three squirrels with all of the points on the perimeter (as we did on page 17). For these polygons, calculate the animal’s home range and range length. For the range length, indicate which two points you are using. Compare these calculations to the ones you made when you disregarded the interior points. What stays the same? What is different? Explain. 2. (A true story) A rancher wrote a local university to pose this problem: “On my land is a water tower that has a diameter of 20 feet. To it, I have tied a cow with a rope that is 20 feet long to a hook fixed on the side of the water tower. What is the area of grass that the cow can graze?” Using a compass, draw a picture on graph paper of the water tower and the cow’s grazing area. Make sure the grazing area is physically possible for the cow to reach. Now estimate the area that the cow can graze. Explain your method of estimating the area. 3. (Another true story) Two mammalogists were capturing bats that originated from one of two caves. The caves were located 500 meters apart. If the bats forage in a circular region that has a center at the cave and that has a radius of 3 kilometers, what is the area of the overlap of the foraging area of the two caves? Use graph paper and a compass to graph the regions for the two caves and use this to estimate or compute the area of the overlapping region. Explain your method of estimating the area. How does the size of the area of the overlapping region compare to the size of the area of one of the circles? 4. Energy can be stored in springs and released as the spring uncoils. This mechanism is commonly found in mechanical clocks that are wound. The more the spring is compressed, the more energy it holds; however, due to the nature of springs, it takes more effort (or force) the further the spring is to be compressed (See Table 2.3). The units for force are Newtons (N) where 1 Newton is the amount of force needed to accelerate a mass of 1 kilogram 1 meter per second per second. The energy stored inside the spring is represented by the area beneath the curve on a graph of the compressional force versus the distance com- 32 CHAPTER 2. GEOMETRY

pressed. The following data gives a relationship between the distance the spring is compressed and the eﬀort needed to compress the spring.

distance compressed (m) Force required to compress (N) 0.5 10 1.0 20 1.5 30 2.0 40 2.5 45 3.0 50 3.5 55 4.0 60

Table 2.3: Displacement versus compressional force

Plotting these points with the x axis denoting the distance compressed and the y axis denoting the force,− or the eﬀort used to compress the spring, and− using a graph that best ﬁts the points gives the graph 2.1.

0 1 2 3 4 5

Figure 2.1: Displacement versus compressional force

To find how much energy is stored inside the spring when it is compressed to 1 meter, you need to find the area under the curve from x = 0 to x =1. Using this idea, find the amount of energy stored inside the spring when it is compressed 1 meter, 2 meters, and 4 meters. Explain how you calculated the area and make sure to express your answers in the correct units. 2.6. HOMEWORK PROJECTS: 33

5. Estimate the volume and surface area of one of your arms and one of your fingers. You can estimate the volume by estimating the volume of different segments and then adding these segments up. For example, you can estimate the volume of your finger by using a cylinder. The height is easy to measure. It is harder to measure the radius of your finger, but easy to measure the circumference, C =2πr. To find the radius, take the circumference and divide by 2π. Now you can estimate the volume of your finger. Explain what segments you estimated and how you used these to arrive at the total volume and surface area of your arm and finger. Find the ratio of the volume to surface area of your arm and the ratio of the volume to surface area of your finger. Compare these two ratios. What observations can you make about these ratios?

6. Engine manufacturers seek to make their engines as eﬃcient as possible. For combustion type engines, the amount of work done is related to the pressure inside the cylinders in the engine and the volume of the enclosed region inside the cylinders. The cycle for a typical engine is shown in the ﬁgure below:

The amount of work done by the engine is the area enclosed by the region and is usually measured in Joules (1 Joule of energy is a Newton-meter, denoted N m). The optimal amount of work the engine can do is determined by the· area between the top curve and the Volume axis. The ratio of these two areas is the eﬃciency of the engine. a. Estimate the area of the enclosed region. Explain your method of estimation and include a picture of your estimation. 34 CHAPTER 2. GEOMETRY

b. Estimate the area between the Volume axis and the top curve. Explain your method of estimation and include a picture of your estimation. c. Using your answers to a and b, find the efficiency for this engine. d. What should be done to the value for the pressure along the bottom curve to increase the efficiency of the engine? Explain. 7. In Western Europe, rather than patrol the area for speeding, the gov- ernments require that commercial vehicles have a recording speedometer installed. While the engine is running, the speed and time is recorded. The speed recordings are collected by customs officials at various check- points throughout Europe. From the speed information the officials can determine if the driver has been on the road too long or if any speed laws have been broken. Appropriate fines are then imposed. The information on the graph below is taken from one of these recording instruments. Assuming the data is from a single trip on the same road, how far has the truck driven? Explain your method of estimating this. What is the elapsed time? What is the average speed for the truck? If the fine for speeding is 7 Euro for each tenth of an hour of speed in excess of 100 km/h what is the fine for this driver? 2.6. HOMEWORK PROJECTS: 35

8. You have four cubes. They have lengths of 2 meters, 4 meters, 3 meters, and 9 meters, respectively. Calculate the volume and surface area for each cube. Find a relationship between the volume of the first cube and the volume of the second cube. Now find a relationship between the volume of the third cube and the volume of the fourth cube. Find a relationship between the surface area of the first cube and the surface area of the second cube. Now find a relationship between the surface area of the third cube and the surface area of the fourth cube. If a cube having side length x has volume V and surface area S, what would be the volume and surface area of a cube having side length x2 (in terms of V and S, respectively)? 9. Aerobic fitness can be determined in a laboratory test that measures oxygen consumption using a treadmill or stationary bicycle. Common scores of oxygen consumption range from 3 to 4 liters of oxygen per minute (L/min), with endurance athletes reporting 5 to 6 L/min. Suppose you work out on a treadmill for 16 minutes and the relationship between your time spent on the treadmill and your rate of oxygen consumption is shown on the graph below. Estimate the total oxygen you consumed in liters during your first 8 minutes on the treadmill. Estimate the total oxygen consumed in liters during your second 8 minutes on the treadmill. Make sure to explain your method of estimation for both answers. How do these two values compare? Is this what you would expect? What is your average rate of oxygen consumption in liters per minute over the entire 16 minutes? 36 CHAPTER 2. GEOMETRY

10. Data for a male (M) and a female (F) ground squirrel were collected on a 24 24 station live-trap grid. The spacing between traps in each column and× each row was 12 meters. The data from this CMR method is shown in Table 2.4. Use the data for the first 13 days to diagram the home range for each squirrel. Then find the home range size using the minimum area method and the range length as we did in the demonstration. Indicate which two points you used to find the range length. On a separate sheet of paper, repeat the exercise above using all 25 days worth of data. Compare the calculations using 13 days worth of data versus using all 25 days worth of data. Did the range size and range length change considerably? Would you expect it to? Explain

Day Female Male Day Female Male 1 D6 T3 14 D7 O1 2 F6 V2 15 D6 P6 3 F6 S2 16 D7 O6 4 A9 S1 17 B9 U2 5 E5 S3 18 D9 V2 6 E6 P2 19 A9 P11 7 D5 O1 20 A8 O4 8 D6 S7 21 A10 O6 9 A11 Q1 22 B9 S2 10 D7 U2 23 B8 V5 11 D10 P9 24 C7 V4 12 D2 R5 25 C5 O9 13 C6 P7

Table 2.4: Range data for ground squirrels

11. The purpose of this project will be to compare the cost per square inch of several different sizes and/or brands of pizza. You can either order pizzas from a pizza restaurant (ordering at least three different sizes) or can instead buy frozen pizzas from a grocery store (again buying at least 3 different sizes). Indicate where you got your pizzas and their sizes. Calculate the area of each pizza in square inches and also the cost (in cents) per square inch for each of the pizzas. Explain which of the pizzas is the best deal. Now suppose you have a dollar off coupon for each of the pizzas. Recalcu- late the cost per square inch for each of the pizzas if the coupon is used. Did this change the outcome, that is, is the same pizza as before still the best buy? Explain. If you so choose, you may also comment on which pizzas tasted the best and which you will most likely buy in the future. 2.6. HOMEWORK PROJECTS: 37

12. The space shuttle travels at 28,000 KPH (kilometers per hour).

(a) What is the speed of the space shuttle in MPH? (b) What is the speed of the space shuttle in miles per second? (c) At that speed, about how long (in seconds) would it take the space shuttle to travel 50 miles? (d) How long (in minutes) would it take the space shuttle to orbit the earth one time? (Note: you will need to find the circumference of the earth to answer this question.) (e) How long (to the nearest day) would it have taken the space shuttle to fly to Mars if it had ever flown to Mars? (Note: you will also need to find the distance from Earth to Mars to answer this question. The distance from Earth to Mars changes quite a bit, so you can either use the closest it has been to Earth or you can use the average distance that it is from Earth. Indicate which figure for distance you used.)

13. In urban planning care is given to ensure that there is enough ”sitting space” in public spaces. Using the formula that there be one linear foot of sitting space per 100 square feet of open area in public spaces and the Cameron University Campus Master Plan 2025 Map found at https://www.cameron.edu/CMP2025, calculate the amount of sitting space in linear feet needed in the green areas bounded by Gore Boulevard, S.W. 38th Street, Dr. Elsie Hamm Drive, and S.W. 27th Street. Make sure to explain in detail how you got your estimate of square feet of open areas from the map. You will need to determine the appropriate scale using the fact that a football ﬁeld is 120 yards in length. 38 CHAPTER 2. GEOMETRY Chapter 3

Lines

3.1 Objectives:

After completing this chapter, you should be able to

Find, if possible, the slope, x intercept, and y intercept of a line • − − Find the equation of a line • Graph lines • Use linear regression to ﬁt lines to data points • Interpret results from linear data and linear graphs •

3.2 Motivation:

Lines describe relationships between two variables in which the rate of change of one variable with respect to the other variable remains constant. Many applications and situations in a wide variety of fields exhibit linear or almost linear behavior. For example, in business, costs, revenues, and profits can be modeled by lines. Examples of phenomena in the physical sciences that exhibit linear behavior are the temperature and volume of gas under constant pressure, the pressure versus the temperature and the pressure versus the altitude in the atmosphere, the force exerted on compressing or stretching a spring, and the resistance to the flow of electricity along a wire versus the length of the wire. In the biological sciences, lines are used to analyze trends in populations, especially for endangered and threatened species, to estimate time of concep- tion in animals, to design and develop protected areas and refuges, to predict species diversity of bird communities, and to develop habitat suitability indices to identify critical habitats for species.

39 40 CHAPTER 3. LINES

In this chapter, we will explore the use of lines and linear regression in the sciences. We begin, in the next section, to explain how lines can be used to estimate population size.

3.3 Introduction to Population Size Estimation:

Wildlife biologists need to know the number of animals in a specific area, or at least an estimate of the number of animals in a given area, to adequately manage a particular species of concern. Depending on the species, a wildlife biologist may wish to increase (rare or endangered species), decrease (pest species), or maintain (game species) a population at a desirable level. The number of animals that occupy an area can be manipulated in three basic ways: (1) population management; (2) habitat management; and (3) people management. For example, the wildlife biologist at the McGee Creek Wildlife Management Area may increase the number of white-tailed deer on the site by translocating new deer to the site (population management), improving habitat conditions by planting food plots (habitat management), or changing the number or types of permits issued (people management). Before any type of manipulation is undertaken, a reliable estimate of the population size in the area should be determined. Several techniques have been developed by field biologists to estimate population size. The most accurate technique is a census, which is a complete count of an entire population. A complete census, where all individuals of a population are counted, is difficult to perform for wildlife, so other methods have been developed by wildlife biologists to estimate population size. Many sampling procedures have been developed to estimate population size for different species of wildlife. No sampling procedure is universal in its application because of the variation in characteristics of animals and the habitats that they use. One technique used to estimate population size requires the re- moval of animals from the population (usually accomplished by using kill traps) on at least two occasions and plotting the number of individuals previously removed from the population on the horizontal or x axis against the number of individuals removed from the population in a given− period on the vertical or y axis. These points are connected with a line, which is then extrapolated to the− horizontal axis in order to find the x intercept. The x intercept is the estimate of the population size. If the same− sampling protocol− is used and the probability of capture remains constant, then all points representing captures during subsequent trapping periods will occur on this line; however, this seldom occurs in nature. For example, a farmer is tired of pigeons eating her peanuts and decides to buy a pigeon trap. On the first day, she catches 75 pigeons and removes them from the population. The next day, she catches 50 pigeons with the trap in the same amount of time. These data are plotted in the figure below. The first data point is (0,75) because no pigeons were previously removed during the first day. The second data point is (75,50) because she removed 75 pigeons previously and removed an additional 50 on the second day. As you can see on the graph, the 3.3. INTRODUCTION TO POPULATION SIZE ESTIMATION: 41 x intercept is (225,0) indicating that the estimation of the population size is 225− pigeons.

If you are studying rare or endangered species, killing them to estimate the size of the population contradicts your overall goal. One method that is used to monitor the number of animals without killing them is called the capture- mark-recapture (CMR) method. In this case, live capture traps are placed in your study area. An animal caught for the first time would be counted and uniquely marked before being released. If marked animals are recaptured during a subsequent trapping period, they are not counted as new individuals when the accumulated prior catch is being calculated. By uniquely marking individuals that have been captured, there is no need to remove them from the population as they can be readily differentiated from new, unmarked animals. In an ideal situation the data points collected by these methods can be represented as a line when graphed on a Cartesian coordinate system. In this case the x intercept represents the population estimate and the y intercept represents− the number caught in the first trapping period. − The reliability of your population estimate increases with additional trapping sessions. However, in virtually all “real world” situations, the probability of capture does not remain constant across all trapping periods. As a result, all the points don’t necessarily fall on a single line. Instead of guessing where the line should be plotted, you can calculate the best fit regression line for your data set using the least squares method to estimate population size. An appropriate technological tool such as a graphing calculator or a computer program should be used to calculate the best fit regression line. The x intercept of this line can be used to estimate the size of the population. − 42 CHAPTER 3. LINES 3.4 Demonstration:

A wildlife biologist would like to estimate the number of fox squirrels on the Atoka Wildlife Management Area (WMA). On the ﬁrst day, the biologist harvests 120 squirrels on the area. At one-week intervals for the rest of the month, using the same amount of eﬀort, the biologist harvests 60, 30, and 15 squirrels during weeks two, three, and four, respectively. The accumulated prior harvest is listed in Table 3.1. Week 1 2 3 4 Number harvested 120 60 30 15 Accumulated prior harvest 0 120 180 210

Table 3.1: Number of fox squirrels harvested from the Atoka WMA

Exploration Question #1: Graph the ordered pairs from Ta- ble 3.1 on an x-y coordinate system where x is the accumulated prior harvest and y is the number harvested. Draw a line through these points. What is the x intercept for this line? What is the y intercept for this line? What− do each of the intercepts represent in− this situation? The wildlife biologist also could estimate the number of fox squirrels on the Atoka WMA by conducting a CMR study. The biologist sets live traps throughout the area and in the ﬁrst week 120 squirrels are captured, marked, and released. The biologist repeats the same trapping eﬀort during weeks two, three, and four and captures 51, 38, and 21 previously unmarked squirrels, respectively. The accumulated prior catch (remember that previously marked animals are not counted on subsequent captures) is listed in Table 3.2.

Week 1 2 3 4 Number of new individuals marked 120 51 38 21 Number previously marked 0 120 171 209

Table 3.2: Number of fox squirrels live trapped on the Atoka WMA

Exploration Question #2: Graph the ordered pairs from Ta- ble 3.2 on an x-y coordinate system where x is the number of animals previously marked and y is the number of new individuals marked. This time your points should not all fall on a straight line. Use your best judgment and draw a straight line that comes closest to all of the points. For your line, what is the x intercept? What is the y intercept? What do each of the intercepts− represent in this situation− and how closely do you think your line estimates the actual situation? 3.5. THE MATHEMATICS: 43 3.5 The Mathematics: 3.5.1 Slope As we saw with the data set from the Atoka WMA, we can attempt to fit a line to two or more data points. It could be, as for the data points in Table 3.1, that the points all fit on one line, or as for the data points in Table 3.2, that the points don’t all fit on one line. Whether or not the points fit on one line depends on the ratio of the difference of the y values to the difference of the x values. − − Written mathematically, for the points (x1, y1) and (x2, y2), we would have the formula y y m = 2 − 1 . x x 2 − 1 This is called the slope of the line. The slope of the line also tells you the inclination of the line. If the slope is positive, the line goes up as you look at it from left to right; if it is negative the line goes down as you look at it from left to right. The bigger the slope in absolute value, the steeper the line. If the slope is 0, the line is horizontal. If there is a 0 in the denominator of the formula for the slope, we say that the slope is undefined. This occurs when the line is vertical. Exercises :

1. Calculate the slope of the lines through the following pairs of points. a. (0, 120) and (120, 60) b. (120, 60) and (180, 30) c. (180, 30) and (210, 15) d. These are the points that you plotted from Table 3.1. What is true about the slopes you calculated in parts a, b, and c? 2. Calculate the slope of the lines through the following pairs of points. a. (0, 120) and (120, 51) b. (120, 51) and (171, 38) c. (171, 38) and (209, 21) d. These are the points that you plotted from Table 3.2. Why are the slopes you calculated diﬀerent for each of a, b, and c?

When points all lie on the same line, it doesn’t matter which two points you choose to calculate the slope - you should always get the same answer. If the points do not lie on the same line, calculating the slopes between diﬀerent pairs of points will give you diﬀerent answers. 44 CHAPTER 3. LINES

3. Calculate the slope of the lines through the following points. Then graph the line for each pair of points. a. (5, 6) and (10, 16) b. (5, 6) and (10, 11) c. (5, 6) and (10, 6) d. (5, 6) and (10, 1) e. (5, 6) and (10, 4) f. (5, 6) and (5, 10)−

For the graphs in problems 4-7, tell whether the given line’s slope is positive, negative, zero, or undeﬁned. 4. 5.

6. 7. 3.5. THE MATHEMATICS: 45

3.5.2 The Equation of a Line The equation that describes a line can be written in a variety of ways. Some of these are Point-Slope: y y = m(x x ) − 1 − 1 Slope-Intercept: y = mx + b General: ax + by = c

In these equations, x and y are variables and m, b, a, c, x1, and y1 are constants (numbers). An important thing to notice is that the x and y are raised to the power of 1 (which we usually don’t write.) All of these forms could describe the same line. The point-slope form, y y1 = m(x x1), is usually used when you are given the slope and a point on the− line. It can− also be used if you are given two points on the line, since you know how to calculate the slope from the previous section. The letter m represents the slope and the letters x1 and y1 represent the x and y coordinate of a known point, respectively. For example, suppose you want to find the equation of the line through (0, 120) and (120, 60). The calculation for the slope is 60 120 60 1 m = − = − = . 120 0 120 −2 − Using either point, we can substitute into the point-slope form of the line: 1 y 120 = (x 0). − −2 − (Here we used the point (0, 120)). Although this is the point-slope form, it isn’t simplified completely. If we distribute (multiply) the 1 , we have − 2 1 1 1 y 120 = x + − (0) = x. − −2 2 −2 Adding 120 to both sides, we have 1 y = x + 120 −2 which is the slope-intercept form of the line. This name comes from the fact that (0, 120) is the y intercept of the line, as we saw in the demonstration. If we move the 1 x term− to the same side as the y we get − 2 1 x + y = 120. 2 Multiplying every term by 2 gives us x +2y = 240 46 CHAPTER 3. LINES which is the usual way of writing the general form. All of these equations describe the same line, just written in different forms.

For a horizontal line, the slope is 0. If we substitute this into the slope- intercept form of the line, we get that y =0x + b, or that y = b. So an equation that just has a y variable and no x variable is the equation of a horizontal line. − −

For example, the graph of the line y = 1 is:

1.5

0.5

-3 -2 -1 1 2 3

-0.5

-1

-1.5

-2

For a vertical line, the slope is undeﬁned, so we cannot use either the slope- intercept form or the point-slope form of a line. If you graph the vertical line, however, you can see that the x value is ﬁxed, while the y value can be anything. Thus the equation of a vertical line will be in the form x = c.

So, the graph of x =2 is 3.5. THE MATHEMATICS: 47

Exercises : Find the equations of the lines described in the following problems. Write your answer in slope-intercept form and general form, where possible. In the general form, rewrite the equation so that none of the constants are fractions. 8. The line has a slope of 5 and goes through the point (3,-4). 9. The line has a slope of -5 and goes through the point (-1,0). 10. The line goes through the points (4,6) and ( 4, 6). − − 11. The line has a slope of 0 and goes through (2,5). 12. The slope of the line is undeﬁned and it goes through 1 , 2 . 2 − 3 13. The line is horizontal and goes through the point (-2,1).

3.5.3 Finding Slopes and Intercepts In the previous section we were given the slope and/or point(s) on a line and were asked to find the equation. Now, given the equation of the line, we need to be able to find the slope and intercepts. To find the slope, rewrite the equation of the line so that the y is by itself on the left hand side. This is the point-slope form of the line, so the number that is the coefficient of x is the slope. For example, if our equation is 3x +2y =6, we would subtract the 3x from both sides to get 2y = 3x +6. − Dividing each term by 2, we have 3 y = − x +3 2 3 so the slope of this line is −2 . If the equation of the line is already written in slope-intercept form, we need only to read off the coefficient of x for the slope. Exercises : Find the slope of the following lines. 14. y =2x +3 15. 3x 2y =3 − 16. 1 y = 4x +12 2 − 17. y = 12 18. x = 5 − 19. x = 2y +7 − 20. 3x +4y 10=0 − 21. x =0 48 CHAPTER 3. LINES

For an x intercept, the y coordinate is 0 since it is on the x axis. To find the x intercept,− substitute in−y = 0 into the equation and solve for− x. The point will always− be of the form (?,0). For the line 3x +2y =6, if we substitute in y =0, we get 3x +2(0) = 6 or 3x =6. Dividing by 3 gives us x = 2 so the x intercept is (2,0). For a y intercept, the x coordinate− is 0 since the point is on the y axis. To find the−y intercept, substitute− in x = 0 into the equation and solve− for y. The point will− always be of the form (0,?). For the line 3x +2y =6 if we substitute in x = 0 we get 3(0)+2y =6 or 2y =6. Dividing by 2 gives us y = 3 so the y intercept is (0,3). − Exercises : Find the x intercept and y intercept of the following lines, if possible. − − 22. y =2x +3 23. 3x 2y =3 − 24. 1 y = 4x +12 2 − 25. y = 12 26. x = 5 − 27. y =7x 28. x = y 2 − 3.5.4 Graphs Once we have information about a line, such as points on the line, the slope of the line, or the equation of the line, we can use this information to graph the line. Since a line is uniquely determined by any two points on the line, we can graph the line if we know two points on the line. Conversely, given the graph of a line, it is possible to find the slope of the line, the intercepts of the line, or arbitrary points on the line. 3.5. THE MATHEMATICS: 49

Exercises : For problems 29-35, graph the line using the given information.

29. The line through (3,4) and (5,6)

30. The line through (3,4) with a slope of 2

1 31. The line through (3,4) with a slope of −3

32. The line whose equation is y =2x +3

33. The line whose equation is 3x 2y =3 − 34. The line whose equation is y = 12

35. The line that is vertical and goes through the point (-2,4)

For the graphs in problems 36-41, ﬁnd, if possible, the slope, the x intercept, and the y intercept. − −

36. 37.

38. 39. 50 CHAPTER 3. LINES

40. 41.

3.5.5 Functions If we think of an equation of a line like y = 2x +3, we can ﬁnd points on the graph of the line by substituting in x values and calculating their corresponding y values. For example, we can substitute− x = 0 into the equation y =2x +3 to see− that the corresponding y value is y =2(0)+3=3. So the point (0,3) is on the graph of the line. In this− case x is called the independent variable since we can pick whatever value for x we would like. Since the value of y depends on the value we have picked for x, y is called the dependent variable. If each value for x determines a unique value of y, then y is said to be a function of x. Any line that can be written in the form y = mx + b is a function of x. Notationally, it is sometimes convenient to write f(x) instead of y when y is a function of x. So for our example of y =2x+3 we would write this in function notation as f(x)=2x+3. The advantage to this notation comes in substituting values for x. For example, to show that we are going to substitute in 0 for x, we would write f(0). For our example, f(0) = 2(0)+3 = 3. Likewise, for our example, f(1) = 2(1)+3 =5. We will be studying functions throughout the rest of this book, although in this chapter we are only studying linear functions. For example a quadratic function would be written as f(x) = x2 +4. Although we will discuss these types of functions in more detail in the following chapter, we still evaluate the function in the same way. For example, f(3) = (3)2 +4=13. 3.5. THE MATHEMATICS: 51

Exercises :

42. For the function f(x)=4x 3, find − a. f(0) b. f(1) c. f( 1) − d. f(10) e. f( 5) − 43. For the function g(x) = 3x2 +1, find − a. g(0) b. g(1) c. g( 1) − d. g(10) e. g( 5) − 44. There is one type of line whose equation can not be written in the form y = mx + b. What kind of line is this? Why does this line not satisfy the definition of a function? 45. Explain why y is not a function of x in the equation y2 =4 x2. − 52 CHAPTER 3. LINES

3.5.6 Linear Regression As mentioned in Section 3.3, “real world” data points do not usually fall on a single line. When plotted on an xy plane, however, there may be at least a linear pattern. For example, consider− the following scatterplots:

16 16

14 14

12 12

10 10

8 8

6 6

4 4

2 2

2 4 6 8 10 2 4 6 8 10

Scatterplot 1 Scatterplot 2

16 16

14 14

12 12

10 10

8 8

6 6

4 4

2 2

2 4 6 8 10 2 4 6 8 10

Scatterplot 3 Scatterplot 4 3.5. THE MATHEMATICS: 53

In the first scatterplot, even though the data points are not all exactly on a single line, there is a very strong linear pattern. The x values and y values increase and decrease together (i.e. large x values are− accompanied by− large y values.) This means that the line that best− fits the data set would have a positive− slope. When this happens, we say that there is a strong positive linear correlation between x and y. In the second scatterplot there is still a strong linear correlation but this time y decreases as x increases and vice-versa. In this case we say that there is a strong negative linear correlation between x and y because the “best fit” line has a negative slope. In the third scatterplot, the linear pattern is not as strong as in the first two scatterplots. However, one can still envision a distinct line with positive slope that passes through the middle of the data points. Here we say that there is a moderate positive linear correlation between x and y. Finally, in the fourth scatterplot, the data points seem to be scattered at random. No single line seems to fit the data set best. In this situation, we would say that there is little or no linear correlation between x and y. Once we determine that our variables are linearly correlated, the next step is to find the equation of our “best fit” line. Several methods exist for doing this. One of the more common methods is called the least-squares method. In this method we consider the “best-fit” line to be the line that minimizes the squared vertical distances between the line and each of the data points. While this method requires calculus and is beyond the scope of this text, many software packages or graphing calculators can calculate this line for us. Keep in mind that the “best-fit” line is not always a good fit. The least- squares method could be used on the data set in the fourth scatterplot above. Even though the computer would give us the equation of a line, that line would not really fit the data points. In addition to giving us the equation of a line, the computer also computes an R2 value. This R2 value is a measure of how well the line fits our data set. If R2 = 0, then there is no linear correlation at all between our two variables and the line does not fit our data set very well. If R2 =1, then all of our data points fall on exactly the same line and we have a perfect fit. Usually R2 is somewhere between 0 and 1. If it is close to one, then our variables are highly correlated and the line fits our data set well. If it is close to 0 then there is little linear correlation between our variables and the line does not fit our data set well. For the four original scatterplots, the R2 values are respectively, 0.98, 0.95, 0.58, and 0.01. 54 CHAPTER 3. LINES

The variable being predicted is called the dependent variable (y) and the one we are using to do the prediction is called the independent variable (x). For the data set in Table 3.2 the number of previously marked animals is the independent variable and the number of new individuals is the dependent variable in the following scatterplot. The R2 value for this plot is 0.98.

120

100

50 100 150 200

Next, we ﬁt a line to the data points from Table 3.2. The equation of this line is y = 116.636 0.47309x. Compare this to the line that you drew by hand for Exploration− Question #2. How closely does your line match the linear regression line?

120

100

50 100 150 200 3.5. THE MATHEMATICS: 55

Exercises : Match the following scatterplots with their corresponding R2 values. 46. R2 =1 47. R2 =0.93 48. R2 =0.67 49. R2 =0.114

20 20

18 17.5

15 16 12.5 14 10 12 7.5 10 5 8 2.5

2.5 5 7.5 10 12.5 15 17.5 20 2.5 5 7.5 10 12.5 15 17.5 20

Scatterplot A Scatterplot B

20 20

17.5 17.5

15 15

12.5 12.5

10 10

7.5 7.5

5 5

2.5 2.5

2.5 5 7.5 10 12.5 15 17.5 20 2.5 5 7.5 10 12.5 15 17.5 20

Scatterplot C Scatterplot D 56 CHAPTER 3. LINES

Exercises : For the following pairs of variables, tell whether you would expect them to have a positive linear correlation, a negative linear correlation, or no linear correlation. Explain your reasoning.

50. Weight and Blood pressure

51. Math ACT score and Grade in Algebra

52. Years smoking and Lung capacity

53. Number of police per 1000 people and Crime Rate

54. IQ and Number of siblings

3.6 Extrapolation and Interpolation:

Once we have ﬁt a least-squares line to the data, we can use the equation of the line to predict values for one of the variables given the other variable. For example, the following data from the U.S. Bureau of the Census, Statistical Abstract of the United States: 1994 in Table 3.3 shows the percentage of the United States population between 18 and 25 years old who smoked in the indicated years.

Year % smokers among 18 to 25 year olds 1974 48.8 1979 42.6 1985 36.8 1988 35.2 1992 31.9

Table 3.3: Percentage of smokers among 18 to 25 year olds

Plotting these points with the year as the independent variable and the percentage of smokers as the dependent variable, we get the equation of the least-squares line to be y = 0.92632x + 1876.4 where x is the year and y is the percentage of 18 to 25− year olds who smoked. The R2 value for this line is 0.98546. Once we have this equation, we can predict the percentage of the population between 18 and 25 who smoked in a certain year. Depending on where the data point is in relation to the given data points we are either interpolating or extrapolating. When we use a data point that falls within our range of values we are interpolating. For this example, using a year between 1974 and 1992 would be interpolating. The higher the R2 value, the more confidence we would place in our prediction. Since the R2 value was 0.98546 for this line, we would be fairly confident in predictions for years between 1974 and 1992. For example, 3.6. EXTRAPOLATION AND INTERPOLATION: 57 substituting 1990 in for x, we would predict the percentage of 18 to 25 year olds who smoked in 1990 to be approximately 33.1%. Remember, however, that this is just a prediction based on the available data. If we use a data point that falls outside our range of values, we are extrapolating. For this example, if we use a year before 1974 or after 1992, we would be extrapolating. For example, we could substitute in 1997 for x to predict the percentage of 18 to 25 year olds who smoked in 1997. However, since we are using data gathered before 1997, we would not be as confident in our prediction, even though the R2 value is high. The equation predicts that the percentage of the 18 to 25 year olds who were smoking in 1997 to be approximately 26.6%. However, smoking among 18 to 25 year olds started to increase in 1995 so this prediction is too low. The farther your prediction point is from the original data, the less confidence you should place in the prediction. Another problem with extrapolating is that it can give mathematically correct answers that are meaningless in context. For example, we could use the equation of the above line to predict the percentage of 18 to 25 year olds who smoked in 2026. Substituting in 2026 for x, we have that y .32. As it doesn’t make sense to have a negative percentage of smokers, this ans≈ −wer is meaningless in this context. Exercises :

55. What is the slope of the least-squares line given on the previous page? 56. What is the x intercept for the least-squares line given on the previous page? − 57. In words, what does the x intercept represent in this example? − 58. Is substituting y = 0 into the above equation an example of interpolation or extrapolation? 59. What is the prediction for the percentage of smokers between the ages of 18 to 25 who smoked in 1987? Is this an example of interpolation or extrapolation? 60. What is the prediction for the percentage of smokers between the ages of 18 to 25 who smoked in 2030? Is this an example of interpolation or extrapolation? Does your answer make sense in this context? 58 CHAPTER 3. LINES 3.7 Homework Projects:

For each assigned project, write your answer in complete sentences. Projects will be graded based on thoroughness and neatness as well as accuracy. Make sure to show your work in any calculations that you do.

1. In this project we are going to simulate the CMR method discussed in the demonstration without actually trapping any animals. To do this, you should fill a bucket or hat with 20 pieces of paper, all cut to the same size. For the first data point, pull out a handful of the pieces of paper (without looking.) These are the “animals” that you caught on the first try and will be the number of new individuals marked in your table. Mark these with a pen or pencil, put them back into the hat, and mix them up. Pull out another handful for the second catch. Disregard the marked ones and count the number of unmarked ones. Mark the unmarked papers and put them all back in the hat. Continue this procedure until you have data points for 5 trapping periods. Make a table like Table 3.2 for your data set. Use technology to make a scatterplot for your data and find the “best fit” line. Interpret the R2 value for this data. Find the x intercept and the y intercept for your line. Interpret what these intercepts− mean for the CMR− method.

2. Table 3.4 gives the winning times (in minutes and seconds) in the Olympic women’s 400m freestyle event.

Year Time Year Time 1952 5:12.1 1984 4:07.10 1956 4:54.6 1988 4:03.85 1960 4:50.6 1992 4:07.18 1964 4:43.3 1996 4:07.25 1968 4:31.8 2000 4:05.80 1972 4:19.04 2004 4:05.34 1976 4:09.89 2008 4:03.22 1980 4:08.76 2012 4:01.45

Table 3.4: Winning times in the Olympic women’s 400m freestyle event

Use technology to make a scatterplot of this data set with the year as the independent variable and the winning time in seconds for the dependent variable. Then ﬁt this data set with a least-squares line. Interpret the R2 value for this line. Using your linear model, predict the winning time in minutes and seconds in the 2016 Olympics and in the 2096 Olympics. According to your model, predict when a woman will be able to swim the 400m freestyle race in 0 seconds. Explain the reasonableness of your model. 3.7. HOMEWORK PROJECTS: 59

3. To get into medical school, students are required to take a standardized test called the Medical College Admissions Test, or MCAT for short. Ta- ble 3.5 gives the GPA and the MCAT scores of students who have taken the MCAT. Before you begin analyzing the data, begin your project report by predicting whether or not GPA and the MCAT scores are highly correlated. Explain why you might expect this. Using technology produce a scatterplot for this data set with the GPA as your independent variable and the MCAT scores as your dependent variable and ﬁnd the “best ﬁt” line. Interpret the R2 value for this line. Does the data support your hypothesis that you made about the MCAT and GPA? Explain. If your GPA is a 3.60, what MCAT score does this model predict that you will get? How reliable is this prediction? Explain.

GPA MCAT GPA MCAT GPA MCAT GPA MCAT 4 18 4 22 3.95 18 3.25 25 3.92 19 3.92 33 3.83 27 3.24 36 3.83 25 3.79 41 3.77 13 3.14 43 3.77 26 3.65 12 3.63 32 3.14 18 3.62 36 3.58 21 3.58 35 3.12 25 3.54 27 3.54 34 3.53 37 3.09 35 3.51 18 3.51 40 3.45 26 3.05 44 3.45 28 3.44 35 3.43 22 3.04 15 3.42 38 3.42 29 3.4 20 3.02 24 3.4 39 3.4 26 3.4 19 2.97 35 3.32 26 3.32 30 3.32 15 2.97 20 3.32 44 3.29 41 3.29 23 2.97 16 3.29 16 3.25 42 3.25 34 2.96 24 2.95 10 2.95 12 2.86 14 2.83 26 2.53 27 2.53 10 2.53 19 2.53 34

Table 3.5: GPA and MCAT scores

4. Determine the shoe size and height of 8 to 10 adult males. Before you begin analyzing the data, begin your project report by predicting whether or not shoe size and height are highly correlated. Explain why you might expect this. Using technology produce a scatterplot for your data set and ﬁnd the “best ﬁt” line. Interpret the R2 value for this line. Does the data support your hypothesis that you made about shoe size and height? Explain. Use the equation of the least-squares line to predict the height of Kevin Durant given that he wears a size 16 shoe.

How close is your prediction to Kevin Durant’s actual height? Explain what factors might have inﬂuenced the reliability of your prediction. 60 CHAPTER 3. LINES

5. Determine the shoe size and height of 8 to 10 adult females. Before you begin analyzing the data, begin your project report by predicting whether or not shoe size and height are highly correlated. Explain why you might expect this. Using technology produce a scatterplot for your data set and find the “best fit” line. Interpret the R2 value for this line. Does the data support your hypothesis that you made about shoe size and height? Explain. Use the equation of the least-squares line to predict the height of Angelina Jolie given that she wears a size 9 shoe. How close is your prediction to Angeline Jolie’s actual height? Explain what factors might have influenced the reliability of your prediction. 6. The United States Standard Atmosphere measures the average temperature and pressure of the atmosphere at a given altitude. The information displayed in the graph below is for the pressure (measured in millibars or mbs) and the temperature in degrees Celsius. Determine the equation of the line that best fits the data points. Using that equation, what is the temperature at a pressure of 500 mb? What is the pressure at a temperature of five degrees Celsius? Verify your results on the graph by graphing the two points you have just found on the line. Explain in words what information this graph is conveying. 3.7. HOMEWORK PROJECTS: 61

7. Determine the weight and height of 8 to 10 adult males. Before you begin analyzing the data, begin your project report by predicting whether or not weight and height are highly correlated. Explain why you might expect this. Using technology produce a scatterplot for your data set and find the “best fit” line. Interpret the R2 value for this line. Does the data support your hypothesis that you made about weight and height? Explain. Use the equation of the least-squares line to predict the height of Kevin Durant given that he weighs 235 lbs. How close is your prediction to Kevin Durant’s actual height?≈ Explain what factors might have influenced the reliability of your prediction. 8. The United States Standard Atmosphere measures the average temperature and pressure of the atmosphere at a given altitude. The information displayed in the graph below is for the altitude and the pressure. Deter- mine the equation of the line that best fits the data points. Using that equation, what is the pressure at altitudes of 3 km and at 7 km? At what altitude would you expect the pressure to be 500 mb? Verify your results on the graph by graphing the three points you have just found on the line. Explain in words what information this graph is conveying.

9. In this project we are going to study the relationship between temperature and humidity. For your home town, ﬁnd the hourly temperature and humidity readings from approximately 7 a.m. until 11 p.m. for some day this month. Make sure you pick a day where there is some variability in the humidity. You should be able to ﬁnd the information at this website: http://www.wunderground.com/history/. For some cities, the readings 62 CHAPTER 3. LINES

may be given on the half hour or quarter hour - if so, adjust the time periods accordingly (e.g. 7:30 a.m. to 11:30 p.m. or 7:15 a.m. to 11:15 p.m.). If you are unable to ﬁnd the information for your home town then use a nearby city instead. In any case, make sure to indicate which city and which day you are using. Use technology to make a scatterplot of this data set with the temperature as the independent variable and the humidity as the dependent variable. Then ﬁt this data set with a least-squares line. Interpret the R2 value for this line. Using your linear model, predict the humidity for temperatures of 83 and 60. Discuss the reasonableness of your answers.

10. In this project we are going to study temperature variation in Oklahoma using the Mesonet. Using the Mesonet website found at http://www.mesonet.org search for the monthly summaries to ﬁnd the maximum air temperature for a selected station for the month of November. Make sure to indicate which station and which year you are using. Use technology to make a scatterplot of this data set with the day as the independent variable and the temperature as the dependent variable. Then ﬁt this data set with a least-squares line. Interpret the R2 value for this line. Using your linear model, predict the temperature for December 15th and for November 15th. Discuss the reasonableness of your answers.

11. The United States Bureau of the Census collects data on highest level of education obtained by gender and median income. Table 3.6 lists data from 2010 where 12 years of school indicates a high school graduate, 16 years of school indicates completion of a bachelor’s degree, 18 years of school indicates completion of a master’s degree, 20 years of school indicates completion of a professional degree, and 22 years of school indicates completion of a doctoral degree. Figures are for people who are 25 years or older.

Men Women Years of School Median Income Median Income 10 21,950 15,650 12 32,501 22,452 13 39,738 26,615 14 42,348 31,537 16 57,131 40,572 18 71,897 51,099 20 101,471 65,788 22 92,419 72,057

Table 3.6: Income levels relative to years of school and gender 3.7. HOMEWORK PROJECTS: 63

Use technology to make a scatterplot of this data set with the years of school as the independent variable and the income for men as the dependent variable. Then fit this data set with a least-squares line. Interpret the R2 value for this line. Using your linear model, predict the income level of a male completing an associate degree. Also, predict the income level of a male who drops out of high school after the 9th grade. Explain the reasonableness of this model. Repeat the above for women with the years of school as the independent variable and the income for women as the dependent variable. Compare the two models and indicate whether years of education is a better pre- dictor of salary for males or females. 12. In order to become aerobically fit, the Center for Disease Control (CDC) recommends exercising for either 2 hours and 30 minutes at moderate intensity OR 1 hour and 15 minutes a week at vigorous intensity (see http://www.cdc.gov/physicalactivity/everyone/guidelines/adults.html.) To receive the maximum benefits from your exercise, you should exercise within what is called the target heart rate zone. Individuals who are sedentary or who have been exercising regularly for less than six months, are considered beginners or intermediates. To calculate your target heart rate (THR) in this category, we use the formula:

THR = MHR PEI × where MHR is your maximum heart rate and PEI is the percent of your exercise intensity. Your maximum heart rate can be calculated by taking 220 minus your age. Calculate your maximum heart rate and plug it into the above formula. For your maximum heart rate, graph the equation where PEI is the independent variable written as a decimal and can range from 0 to 1. If your target heart zone is for workouts where the percent exercise intensity ranges from 60% to 80%, what is your minimum target heart rate and your maximum target heart rate? Indicate the target heart zone on your graph. Individuals who have been exercising for at least six months on a reg- ular basis are considered advanced and calculate their target heart rate diﬀerently. For this category, we use the formula

THR = [(MHR RHR) PEI] + RHR − × where RHR is your resting heart rate. Calculate your resting heart rate by measuring your heart rate for one minute for three mornings before rising out of bed and taking the average of your measured heart rates. Plug in your MHR and your RHR into the above formula, and graph this new equation where PEI is the independent variable written as a decimal and can range from 0 to 1. The target heart rate zone is still for workouts 64 CHAPTER 3. LINES

where the percent exercise intensity ranges from 60% to 80%. What is your minimum target heart rate using this equation and what is your maximum target heart rate? How do these compare with the minimum and maximum you found using the formula for beginners/intermediates? Which do you think is more accurate for you? Now assume that your resting heart rate stays the same over the next ten years and that you always exercise at a 70% intensity. Plug your RHR, PEI, and the formula for MHR into

THR = [(MHR RHR) PEI] + RHR − × and simplify this equation. Graph the simplified equation where your age is the independent variable and ranges from your current age to 10 years past your current age. What does this graph tell you about your target heart rate as you get older? 13. The size of an animal’s home range is related to how much the animal weighs. In Table 3.7 (on the next page) the ranges of body weight and home range size of several species of carnivores are listed. For the females, use grams for the weight and hectares for the home range size. Using technology make a scatterplot for females with body weight as the independent variable and home range size as the dependent variable. Fit a line to the data and interpret the R2 values for the line. Use the relationship between body weight and home range size to estimate the home range size of a female mink (567 g), female bobcat (7 kg), female red wolf (18 kg), and female black bear (90 kg). After you have completed your calculations, use the library or the Internet (or your friendly neighborhood mammalogist) to find the home range size reported in the scientific literature for each of these animals. If your results differ markedly from those reported, discuss reasons that may have led to these differences. 14. The size of an animal’s home range is related to how much the animal weighs. In Table 3.7 (on the next page) the ranges of body weight and home range size of several species of carnivores are listed. For the males, use kilograms for the weight and square kilometers for the home range size. Using technology make a scatterplot for males with body weight as the independent variable and home range size as the dependent variable. Fit a line to the data and interpret the R2 values for the line. Use the relationship between body weight and home range size to estimate the home range size of a male mink (1362 g), male bobcat (16 kg), male red wolf (32 kg), and male black bear (214 kg). After you have completed your calculations, use the library or the Internet (or your friendly neighborhood mammalogist) to find the home range size reported in the scientific literature for each of these animals. If your results differ markedly from those reported, discuss reasons that may have led to these differences. 3.7. HOMEWORK PROJECTS: 65

Species Female Male Female Male Home Body Body Home Range Size Weight Weight Range Size Least 30 g 100 g 0.2 ha 26.2 ha Weasel Long-tailed 70 g 300 g 12 ha 160 ha Weasel Spotted 280 g 1,000 g 64 ha 4,360 ha Skunk Striped 1.2 kg 5.3 kg 234 ha 512 ha Skunk Gray Fox 3 kg 7 kg 75 ha 653 ha Red Fox 3 kg 7 kg 42 ha 4.6 km2 Badger 8 kg 12 kg 500 ha 850 ha Raccoon 6 kg 11 kg 229 ha 4,950 ha Coyote 8 kg 20 kg 8 km2 42 km2 Mountain 35 kg 65 kg 96 km2 293 km2 Lion Brown 100 kg 675 kg 79 km2 1,398 km2 Bear

Table 3.7: Body Weight versus Home Range Size 66 CHAPTER 3. LINES Chapter 4

Quadratics

4.1 Objectives:

After completing this chapter, you should be able to

Find, if they exist, the vertex, x intercept(s), and y intercept of a parabola • − − Determine if a parabola opens up or down and ﬁnd the maximum or • minimum value for a parabola

Graph a quadratic equation • Use regression on quadratic data • Understand some of the physical uses for the quadratic equation • Distinguish between linear and quadratic data • 4.2 Motivation:

Although lines may be the easiest equations to work with algebraically, not all data sets are linear in nature. Another type of equation is the quadratic equation, usually written as y = ax2 + bx + c, a = 0. The Babylonians, an ancient civilization that flourished from about 20006 B.C. to 300 A.D., and the Egyptians, whose civilization flourished from 3500 B.C. until conquered by the Greeks in 332 B.C., were able to solve some specific types of quadratic equations. The Greek mathematician Diophantus, who lived around 250 A.D. also was able to solve quadratic equations, but only considered positive rational solutions. The Hindu civilization from 200 A.D. to 1200 A.D. recognized that quadratics have two solutions and included negative and irrational solutions. Although they applied their algebra techniques to business, the main application was astronomy.

67 68 CHAPTER 4. QUADRATICS

Today, quadratic equations describe a wide range of natural phenomena. In the biological sciences, quadratic equations are used to estimate the rate of change of limited population growth. In electronics, quadratics are used in circuits, and continuing the Hindu tradition, quadratics are still used to study orbital mechanics. In chemistry, reaction rates and chemical equilibria can be described using quadratic equations. In physics, quadratic equations are used to describe the motion of projectiles (kinematics). In the next section, we will begin to explore how quadratics are used to describe the motion of a falling object.

4.3 Introduction to Kinematics:

Kinematics is the study of the motion of objects. The goal of kinematics is to construct a set of relationships so that given any set of initial conditions for the system, predictions about the system in the future may be made. The basic quantities that are of interest are the position, velocity, and acceleration of the object as well as the time that the object has been in motion. Position is the object’s location in space relative to an arbitrary point called the origin. Velocity is a measure of how fast the object changes position and in what direction. If you are only measuring how fast the object changes position, then your measurement is speed. In the metric system, the units for velocity are usually given in meters per second. Acceleration is a measure of how fast the velocity changes. In the metric system, the units for acceleration are usually given in meters per square second. One way to measure velocity is by computing the average velocity (v). We usually compute the velocity as the ratio of the distance covered to the elapsed time. If y denotes the ﬁnal position, y0 denotes the initial position, and t denotes the elapsed time, this formula is given by y y v = − 0 . t If we multiply both sides by time, we get the more familiar relationship

y y = vt, − 0 or “distance equals rate times time”. Another method for calculating the average velocity is to average the velocity at the beginning (v0) and end (v) of the interval of interest. This method gives us the formula v + v v = 0 . 2 Similarly, average acceleration (a) is defined as the ratio of the difference between the final velocity, v, and the initial velocity, v0, and the elapsed time, t. This formula is given by v v a = − 0 . t 4.4. DEMONSTRATION: 69

We can combine these three formulas to get an equation that describes the location of an object at any time t, if we know it has acceleration a, initial 1 2 velocity v0, and initial position y0. This equation is y = 2 at +v0t+y0, which is a quadratic equation in t. In the next section, we will actually measure the time it takes for an object to fall from a certain height and construct a quadratic equation to describe this motion.

4.4 Demonstration:

The goal of this demonstration is to measure the relationship between how far an object falls and the time it takes for the object to fall. Using an apparatus that measures how long it takes for a ball to fall from a preset height, ﬁll in the following table with diﬀerent heights and times. Measure the height in meters and the time in seconds.

time height of ball

Plot this data set on a graph with the time on the x axis and the height of the ball on the y axis. Make sure to use the same scale− on both the x and y axes. −

Exploration Question #1: Draw a smooth curve that is closest to all of the points. To determine if this graph is linear, we can calculate the slopes between diﬀerent points and compare. Calculate the slopes between consecutive pairs of points. (There will be nine slopes to calculate.) Are the slopes the same? Can you decide if the data set is linear or not from the slopes?

The shape of your graph should be a portion of a parabola. The general equation for a quadratic equation in this situation is usually written as 1 y(t) = gt2 + v t + y 2 0 0 where g is the constant due to gravity, t is the elapsed time, v0 is the initial velocity, and y0 is the height of the ball that hits the ground after 0 seconds. For 70 CHAPTER 4. QUADRATICS

our experiment, v0 = 0 since the ball was dropped, not thrown. This reduces the equation to 1 y(t) = gt2 + y . 2 0 When the ball was dropped in the experiment, we were ﬁnding the time it took for the ball to hit the ground. Since it takes 0 seconds for a ball to drop 0 meters, we have y(0) = y0 =0. The above equation then becomes

1 y(t) = gt2. 2 Exploration Question #2: Substitute the data points that you collected for the time and the height into the above equation and solve for g. Do this for each pair of data points, and then average the values that you get for g. How close is your average to 9.81? The accepted value for g, the acceleration due to gravity on earth, is 9.81 m/sec2. Why would your values for g not be exactly 9.81 m/sec2?

4.5 The Mathematics: 4.5.1 Vertex A quadratic equation is an equation which can be written in the form

y = ax2 + bx + c where a =0. Since each value for x determines a unique value for y, quadratic equations6 of this form are also functions and we can write them using function notation as f(x) = ax2 + bx + c. As we saw in the demonstration, one use for the quadratic equation is to describe the position of an object dropped from a given height. In that example, we used t for the variable time, and in general, the letter we use for the variable will change depending on the context. The quadratic equation has many other applications as well, and it will be useful to know the following information about a quadratic equation. The shape of the graph of a quadratic equation is called a parabola. It can either open up or down and be narrow or wide. If a is negative, the parabola opens down; if a is positive the parabola opens up. The position of the parabola also depends on its x and y intercepts and its vertex. The vertex of the parabola is the “turning point.” If− the parabola opens up, the vertex is the point that has the lowest (or minimum) y value; if it opens down, the vertex is the point that has the highest (or maximum)− y value. The x value of the vertex can be found from the formula − − b x = − , 2a 4.5. THE MATHEMATICS: 71 where a is the coeﬃcient for the x2 term and b is the coeﬃcient of the x term. For example, if the quadratic equation− is −

y = x2 +4x +3, the x coordinate for this vertex would be − b 4 x = − = − = 2. 2a 2(1) −

Once you have the x coordinate, you can substitute this value back into the quadratic equation to− ﬁnd the y value for the vertex. For this example, if we substitute x = 2 into f(x) = x2−+4x + 3 we have −

f( 2)=( 2)2 + 4( 2)+3= 1. − − − −

The vertex for this parabola is then ( 2, 1). Since the a value for this parabola is positive, this parabola opens up, so− the− minimum y −value for this quadratic is -1. −

Exercises : For problems 1 - 5, ﬁnd the vertex of the parabola. Then decide if the y value of the vertex is the minimum or maximum y value of the parabola. − −

1. y =3x2 +12x 5 −

2. f(x) = 3x2 +12x +5 −

3. g(x) = 3x2 12x 5 − − −

4. y = x2 +3x 7 −

5. f(x)=2x2 +4x 10 −

6. For the equation y = ax2 + bx + c to be a parabola, we stipulated that a =0. If a is equal to 0, what will the graph of this equation be? 6 72 CHAPTER 4. QUADRATICS

4.5.2 Intercepts

In the chapter on lines, we discussed how to ﬁnd x intercepts and y intercepts. The general philosophy still holds here: Since− the x intercept− is the point where the graph crosses the x axis, the y-value must− be 0. Likewise, since the y intercept is the point where− the graph crosses the y axis the x- value must− be 0. It is easiest to ﬁnd the y intercept. − − For the quadratic

f(x) = x2 +4x +3 we can substitute in x = 0 to ﬁnd the y intercept. In this case we would get −

f(0) = (0)2 +4(0)+3=3 so the y intercept is (0,3). − Finding the x intercept involves more work. To ﬁnd the x intercept, we need to substitute− in y =0. When we do this we get −

0 = x2 +4x +3.

There are two common ways of solving a quadratic which is equal to 0 (the fact that it is equal to 0 is important). One is factoring and the other is the quadratic formula.

Factoring: It may or may not be possible to factor a quadratic. If it can be factored, we can then set each factor equal to 0 and solve for x. For example, x2 +4x +3 factors into

(x + 3)(x + 1).

Since the only way you can multiply two things together and get 0 is if one of them is 0, we must have either x +3=0or x +1=0. In the ﬁrst case, x = 3 and in the second case x = 1. So, this quadratic has two x intercepts (-3,0)− and (-1,0). − − 4.5. THE MATHEMATICS: 73

Quadratic Formula: It may not always be possible to factor a quadratic with integers, even if there are solutions to the equation y = 0. In this case, the quadratic formula can be used to ﬁnd the values for x. It is

b √b2 4ac x = − ± − . 2a The symbol means that there are actually two formulas here: one with a plus and one± with a minus. For example, to solve the quadratic

x2 +6x +3=0 we could ﬁrst try factoring, but would not be successful. Using the quadra-tic formula, we would have

6 62 4(1)(3) 6 √36 12 6 √24 x = − ± − = − ± − = − ± . p2(1) 2 2

To simplify this further, we need to simplify the square root by noticing that 24=4 6. We would then have · 6 √4√6 6 2√6 2( 3 √6) x = − ± = − ± = − ± = 3 √6. 2 2 2 − ±

There are two solutions here: One is 3 + √6 which is approximately -.55 and the other is 3 √6 which is approximately− -5.45. − − It may be that the number under the square root is negative. In this case, we would say that there are no real solutions, or in the case of intercepts, that there are no x intercepts. − Exercises : For problems 7 - 12, ﬁnd, if possible, the x and y intercepts of the given parabolas. −

7. y = x2 +3x 4 − 8. f(x) = x2 +3x +5

9. g(x) = 3x2 12x 5 − − − 10. y =2x2 7x 15 − − 11. y = 2x2 +4x 5 − − 12. f(x)=3x2 6x +1 − 74 CHAPTER 4. QUADRATICS

4.5.3 Graphs

As we discussed in Section 4.5.1, the graph of a quadratic equation is called a parabola. Depending on whether the parabola opens up or down, its basic shape is

The important points for graphing are the vertex and the intercepts. Occa- sionally it might be necessary to plot a few other points to determine the width of the graph, but usually the vertex and intercepts will suﬃce. For example, we determined in Section 4.5.1 that the vertex of the parabola given by the equation

y = x2 +4x +3

is (-2,-1). 4.5. THE MATHEMATICS: 75

In Section 4.5.2, we determined that the y intercept for this parabola is (0,3) and that the x intercepts are (-3,0) and (-1,0).− Combining this information, we have the following− graph:

-5 -4 -3 -2 -1 1

For another example, consider the quadratic equation given by 11 y = x2 3x + . − 4 To ﬁnd the x value of the vertex, we have that − b 3 3 x = − = = . 2a 2(1) 2

To ﬁnd the y value of the vertex, we ﬁnd − 3 3 2 3 11 1 f = 3 + = . 2 2 − 2 4 2

3 1 Thus the vertex is 2 , 2 . 76 CHAPTER 4. QUADRATICS

To ﬁnd the y intercept, we need to ﬁnd f(0) for the y coordinate. Substi- tuting 0 in for x−we have − 11 f(0) = (0)2 3(0) + , − 4

11 so the y intercept is (0, 4 ). To ﬁnd− the x intercept, we need to substitute y = 0 into the equation and solve for x. That− is, we need to solve 11 x2 3x + =0 − 4 by either factoring or using the quadratic formula. In this case, it is probably 11 easiest to use the quadratic formula. Using a = 1, b = 3, and c = 4 and substituting into the quadratic formula we have −

11 3 9 4(1)( 4 ) 3 √ 2 x = ± q − = ± − . 2(1) 2

Since there is a negative under the square root we know that there are no real solutions to this equation. In other words, there are no x intercepts for this parabola. The graph of this parabola, with vertex 3 , 1 ,− y intercept (0, 11 ), 2 2 − 4 and no x intercepts is −

-1 1 2 3 4 4.5. THE MATHEMATICS: 77

Exercises : For problems 13 - 19, graph the following parabolas. Be sure to label the vertex and intercepts on your graph.

13. y = x2 +3x 4 − 14. f(x) = x2 +3x +5 15. g(x) = 3x2 12x 5 − − − 16. y =2x2 7x 15 − − 17. f(x) = x2 8x +3 − − 18. y = x2 4 − 19. y =3x2 6x − 4.5.4 Regression In Section 3.5.6, we learned how to plot real-world data points on a scatterplot to determine if there is a linear relationship between our two variables. If there is a linear pattern, then we can use technology to find the equation of the line that best fit the data set. Sometimes when we make a scatterplot, we may find that our data set is not linear in nature, but instead is quadratic. For example, suppose we want to use a person’s age to predict the amount of money that they earn in a year. The data points in Table 4.1 are from the Bureau of the Census and were collected for the year 2011. They show the age of the head of the household and the income of the householder.

Midpoint of age in category Income of householder 19.5 30,733 29.5 50,199 39.5 61,083 49.5 64,235 59.5 56,973 69.5 31,354

Table 4.1: Income of householders, 2011 78 CHAPTER 4. QUADRATICS

These points were used to make the following scatterplot.

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0 10 20 30 40 50 60 70

Figure 4.1: Age versus Income

If we ﬁt a line to this data set, the result looks like this:

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10 20 30 40 50 60 70

Figure 4.2: Best Linear Fit

While this is the line that “best fits” our data points, it really does not fit very well at all. The R2 value is only 0.009. This means that the two variables are not very linearly correlated. Since the scatterplot resembles a parabola, we might consider fitting a quadratic curve to the data set. To do this, we can fit a polynomial of degree 2. 4.5. THE MATHEMATICS: 79

The result now looks like this:

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20 40 60 80

Figure 4.3: Best Quadratic Fit

The curve now seems to ﬁt the data points much better, and the R2 value is 0.97. The quadratic equation that best ﬁts this data is

Income = 44142.6+ 4812.15Age 53.2159Age2. − − This means that if we want to predict the income of a householder that is 61 years old, we simply plug in 61 for Age to get

Income = 44, 142.6+ 4812.15(61) 53.2159(61)2 = 51, 382.1861. − − Since it really doesn’t make sense for someone to earn 0.1861 of a dollar, we probably would predict that a 61 year old householder would make 51,382.19. However, since we are predicting and all of the values in the tables were≈ rounded to the nearest dollar, we would probably report our answer as 51,382. A word of caution needs to be given here about ﬁtting quadrati≈ cs and higher degree functions to data that is in fact linear in nature. A researcher, when trying to determine the type of relationship between two variables would usually prefer to have a linear relationship rather than a quadratic relationship if possible (the simpler the better). However, for data that does not fall on a perfect line, yet is still linear in nature, a quadratic will always give a higher R2 value than a line. For example consider the following scatterplot: 80 CHAPTER 4. QUADRATICS

17.5

12.5

7.5

2.5

2.5 5 7.5 10 12.5 15 17.5 20

If we ﬁt a line to this scatterplot the R2 value would be 0.9450 and the line would look like this:

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2.5 5 7.5 10 12.5 15 17.5 20 4.5. THE MATHEMATICS: 81

If we ﬁt a quadratic to this scatterplot the R2 value would be 0.9451 and the parabola would look like this:

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2.5

2.5 5 7.5 10 12.5 15 17.5 20

This may look more like the graph of a line rather than the graph of a parabola, but it actually is part of a graph of a quadratic with a really small coefficient of x2. Since the R2 value did not increase significantly from the linear fit to the quadratic fit, we would probably go ahead and use the linear fit for our prediction equation. To help us determine what we mean by “increase significantly”, we want to use the adjusted R2 value which takes into account the degree of the function that we are trying to fit to our scatterplot. It assumes that lower degree equations are preferred if they fit the data just about as well as the higher degree equations. Therefore, it is this value that we consider when trying to choose between prediction equations of differing degrees. For our quadratic fit above, the adjusted R2 value is 0.9294. Since this adjusted R2 value is less than our R2 value for the line we would conclude that the linear equation is the better fit. If the adjusted R2 value is not available we can use the coefficient of the x2 term in the equation of the “best-fit” quadratic to help us determine which equation we should use. For example, the equation of the quadratic for the above scatterplot is

y = 0.00202x2 +0.92086x 0.2637. − − Because the coeﬃcient of the x2 term is close to 0, we can conclude that we should use the linear equation for our model. 82 CHAPTER 4. QUADRATICS

Exercises : For exercises 20 through 23, decide if you should use a line or a quadratic to model the data in the given scatterplot.

400 20

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2.5 5 7.5 10 12.5 15 17.5 20 2.5 5 7.5 10 12.5 15 17.5 20

Exercise 20 Exercise 21

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5 10 15 20 25 30 35 40 2.5 5 7.5 10 12.5 15 17.5 20

Exercise 22 Exercise 23

24. After fitting a line and a quadratic to a scatterplot, you find that the adjusted R2 value for the line is 0.8567, the R2 value for the quadratic is 0.90463, and the adjusted R2 value for the quadratic is 0.87034. Which is the better fit, the quadratic or the line? Explain. 25. After fitting a line and a quadratic to a scatterplot, you find that the adjusted R2 value for the line is 0.9567, the R2 value for the quadratic is 0.96463, and the adjusted R2 value for the quadratic is 0.92034. Which is the better fit, the quadratic or the line? Explain. 4.6. HOMEWORK PROJECTS: 83 4.6 Homework Projects:

1. Enzymes serve as catalysts and speed up the rate of chemical reactions that occur in the cells of organisms. Rates of chemical reaction can be influenced by a variety of parameters such as temperature, concentration of reactant molecules, pH, pressure, and enzymes. In this project, you will investigate the relationship between the rate of product formation and pH in an enzyme-catalyzed reaction. To do this, you will need to use a Spec 20 (spectrophotometer) from either the biology or chemistry labs. The instructor will arrange for you to use this equipment under faculty supervision. You will start with fixed amounts of catechol and catecholase in test tubes exposed to oxygen in the air. However, you will use a different pH buffer in each of the test tubes. You will use a Spec 20 to quantify the amount of product formed during a fixed time interval in each of the test tubes. Using the data points that you collect during this experiment and technology, make a scatterplot with the product formulation as the dependent variable and the pH level as the independent variable. Does the data set look linear or quadratic? Fit a line and a quadratic to this data set and discuss which equation best fits the data set. Find the vertex for the quadratic equation and explain what the vertex represents in this experiment. 2. Area and perimeter share a quadratic relationship. Consider a rectangular field of length L and width W which you wish to fence in using 100 meters of fencing. Using the fact that the 100 meters is the perimeter of the rectangle, solve for W in terms of L and substitute this into the formula for the area of the rectangle (giving you a quadratic in the variable L.) Graph this quadratic on graph paper. For what range of lengths is the area positive? For what lengths is there no enclosed area? (Where are the x intercepts?) For what length is the area greatest? (Where is the vertex− of the parabola?) What are the dimensions of the field with greatest area? What type of rectangle is this? Given an arbitrary perimeter P find a formula (in terms of P ) for the maximum area and the dimensions of the field that can be enclosed? On another piece of property there is already a fence on one side. If you have 100 meters of fencing, what is the maximum area that can be enclosed? What dimensions give you the maximum area? 84 CHAPTER 4. QUADRATICS

3. One way to measure the amount of algae present in a body of water, such as a lake, is to measure the reflectivity of the water surface. This measurement can be much easier to take than actually trying to count the number of algae present in a water sample. To use the reflectivity to predict the amount of algae present we need to have an equation that relates the amount of chlorophyll present in the water and the reflectivity measurement. The following data points in Table 4.2 were collected experimentally. The NIR/RED column is the reflectivity measurement.

chlorophyll NIR/RED chlorophyll NIR/RED 3 1.04 12 1.31 2 1.05 15 1.38 1 1.05 14 1.37 4 1.06 13 1.35 5 1.18 20 1.48 6 1.19 18 1.4 9 1.2 17 1.55 7 1.2 16 1.43 8 1.25 19 1.4 10 1.3 24 1.75 11 1.3 25 1.7 25 1.75 26 1.78 27 1.78 28 1.77 40 1.8 42 1.81 46 1.86 49 1.87 50 1.9 52 1.92 55 1.9 56 1.95 56 1.75 58 1.68 58 1.97 59 2.11 60 1.79 61 2.11 62 2.1 62 2.1

Table 4.2: Ratio of NIR to RED versus chlorophyll

Using technology, make a scatterplot of this data set with NIR/RED as the independent variable and the chlorophyll as the dependent variable. Does it look linear or quadratic? Fit a line and a quadratic to this data set and discuss which equation best fits the data set. For the quadratic equation, find the x intercepts. Explain what the x intercepts are describing in this physical− situation. − At what NIR/RED value would we expect to find the maximum amount of chlorophyll? Explain how you determined this value. 4.6. HOMEWORK PROJECTS: 85

4. Objects moving due to gravity can be described with quadratics. For objects near the surface of the earth the height of the object, h, can be predicted for any time t, given the initial height of the object, h0, and the initial vertical velocity v0. The value for the initial velocity is positive if the object is initially moving upward, negative if it is initially moving downward, and zero if the object is initially at rest. The equation that describes the motion is g h = t2 + v t + h 2 0 0

where g is 9.81 m/sec2, the accepted value for earth’s gravity. − a. If an object is dropped from a height of 20 meters with a zero initial velocity, use the above equation to determine how long it takes for the object to hit the ground. What is the maximum height of the object? b. If instead of dropping the object we throw the object downward at 10 m/sec from a height of 20 meters, use the above equation to determine how long it takes for the object to hit the ground. What is the maximum height of the object? c. If instead the object is thrown upward at 10 m/sec from a height of 20 meters, use the above equation to determine how long it takes for the object to hit the ground. What is the maximum height of the object? At what time does the maximum height occur? d. Graph all three equations with time in seconds on the x axis and height in meters on the y axis. Indicate on your graphs the− maximum heights and when the object− hits the ground.

5. Species diversity is a community-level characteristic. A community in- cludes all the populations of organisms in a given area at some point in time. Species diversity is high when many equally or nearly equally abundant species are present. If a community only has a few species, or if only a few species are abundant, then species diversity is low. Usually, a high species diversity value indicates a complex community with many species interactions. The following data points in Table 4.3 were estimated from Collins and Gibson (See Appendix B). Fire impacts both the number of plant species present and the abundance of each species. The following data were collected to assess the relationship between time in years since last fire and species diversity of plants. Using technology, make a scatterplot of this data set with time as the independent variable and species diversity as the dependent variable. Does it look linear or quadratic? Fit a line and a quadratic to this data set and discuss which equation best fits the data set. For the quadratic equation, find the vertex and the x intercepts. Explain what the vertex is describing in this physical situation.− Are there any values for the independent 86 CHAPTER 4. QUADRATICS

Time Sp. diversity Time Sp. diversity 0 3.9 3 5.6 0 4.1 3 6.9 0 4.2 3 7.5 0 4.5 4 5.9 0 4.7 5 6.2 0 5.0 6 6.8 0 5.1 7 6.6 0 5.2 8 9.9 2 7.2 10 5.8 15 5.4 10 6.6 2 7.1 10 7.5 15 6.4 11 5.2 1 4.6 12 5.3 1 4.7 12 5.4 1 5.3 12 5.8 1 5.8 13 5.2 1 6.1 13 5.3 16 4.7 13 5.9 16 5.7 14 5.2 2 4.1 14 5.7 2 4.3 14 5.8 2 7.1 15 6.4

Table 4.3: Species diversity as related to ﬁre

variable that give unreasonable answers for the dependent variable? Ex- plain. 6. A pendulum is any object that swings back and forth such as a swing on a swing set or your arm as you walk. If the end of the pendulum is significantly heavier than the rest of the pendulum it is called a simple pendulum. The swing on the swing set is a pendulum but it is not a simple pendulum until the end is more massive. This is usually accomplished by sitting on the swing. Any lightweight string with a weight attached will serve as a simple pendulum. Check out a pendulum from your instructor. Watch the pendulum swing. If the pendulum is shortened, how is the motion changed? Measure how much time is required for a specific length pendulum to oscillate one time. Do this for at least ten different lengths. (Note: It will be easier to get an accurate time if you count the time for ten oscillations and divide this time by 10. This helps since it reduces the error caused by starting and stopping the pendulum.) Plot your data using technology with length in meters on the y axis and − 4.6. HOMEWORK PROJECTS: 87

time in seconds on the x axis. Fit a line and a quadratic to your data and discuss which equation− best fits the data. The theoretically predicted equation describing the length of the string in terms of the time is given by g Length = − T ime2, 4π2 where g = 9.81 m/sec2. The number g is the accepted value for earth’s − g gravity. Notice that in this quadratic equation, a = 4−π2 , b = 0, and c = 0. Determine the values of a, b, and c from the quadratic you got using technology. Compare the values you got for a, b, and c with the values from the theoretically predicted equation. Based on your quadratic equation, what value do you predict for earth’s gravity? Discuss reasons for why your experimentally obtained value for g might be different than the theoretically predicted value of 9.81 m/sec2. − 7. Water boils at different temperatures depending on the air pressure. Air pressure is a measure of the weight per unit area of the air above a given location. At sea level the weight of air above a 1 inch square is 14.7 pounds. In the metric system, the weight of air above a 1 meter square is 1.01325 105 Newtons. At higher altitudes, there is less air above any given point× so the air pressure is lower. Thus different altitudes above sea level will have different boiling temperatures. Table 4.4 lists the boiling temperature in degrees Fahrenheit versus altitude in increments of 1000 feet. Thus at sea level, the boiling temperature of water is 212 degrees Fahrenheit and at 10,600 feet above sea level, the boiling temperature of water is 193 degrees Fahrenheit.

Degrees in Fahrenheit Altitude in 1000 feet 249 -15 212 0 193 10.6 179 18.4 115 55 102 63 92 70 45 105

Table 4.4: Boiling temperatures of water

Using technology, make a scatterplot of this data set with the degrees in Fahrenheit as the independent variable and the altitude in increments of 1000 feet as the dependent variable. Fit a line and a quadratic to this data set and discuss which equation best ﬁts the data set. Using the quadratic equation, ﬁnd the predicted boiling point of water at the top of Pikes Peak which has an altitude of 14,110 feet. Find the 88 CHAPTER 4. QUADRATICS

predicted boiling point at the town in which you live. If the boiling point is 200 degrees Fahrenheit, what is the altitude?

8. Johannes Kepler (1571-1630) is credited with one of the giant strides to- ward the relativity theory produced by Isaac Newton in 1687 which describes motion in terms of forces. Kepler published two books on astronomy to describe his ﬁndings based on the observational data of Tycho Brahe; the ﬁrst book was Astronomia Nova (The New Astronomy) published in 1609 and the second book was Harmonices Mundi (Harmony of the Worlds) published in 1619. In these works, Kepler put forth three ideas now regarded as Kepler’s Laws. These laws are 1. the planets circulate around the sun in elliptical orbits with the sun at one focus. 2. the radius vector from the sun to a planet sweeps out equal areas in equal times. 3. the square of the time of one complete revolution of a planet about its orbit (the orbital period) is proportional to the cube of the distance from the planet to the sun. In this project you are going to statistically verify Kepler’s Third Law. In Table 4.5, orbital periods are given in Earth days and Earth years (which consists of 365 days) and the mean orbital distance is given in Astronomical units (AU). One AU is the distance from the Earth to the Sun or 1.5 108 km. × Planet Orbital Period Distance from Sun Mercury 89 days 0.4 AU Earth 365 days 1.00 AU Mars 687 days 1.5 AU Saturn 30 years 9.5 AU Neptune 165 years 30 AU Pluto 248 years 39 AU

Table 4.5: Planetary Orbits and Distance from Sun

Using technology, make a scatterplot with the orbital period in years as the independent variable and the cube of the distance as the dependent variable. Fit a line and a quadratic to this data set and discuss which equation best ﬁts the data set. Using the quadratic equation, predict Venus’ distance from the Sun given that its orbital period is 225 days. If the orbital period of Jupiter is 11.9 years, what would be its predicted distance from the Sun? If the distance from Uranus to the Sun is 19.18 AU, what is the predicted orbital period? Discuss the reasonableness of your answers. 4.6. HOMEWORK PROJECTS: 89

9. The Cornell Nest Box Network began a study in 1997 to study how clutch size (the number of eggs females lay in one nest attempt) varied by geographic location and by year for Eastern Bluebirds. They were interested in studying if clutch size increased from south to north and how the clutch sizes varied from year to year. The data points in Table 4.6 were estimated from the article Clutch-size Variation in Eastern Bluebirds (see Appendix B). Table 4.6 gives the average clutch size of the bluebirds who had laid their eggs by the indicated lay date (the lay date expresses date as days since January 1) at the latitude of 30◦ N. Using technology produce a scatterplot for the data in Table 4.6 with the lay date as the independent variable and the average clutch size as the dependent variable. Fit a line and a quadratic to the scatterplot and discuss which equation bests ﬁt the data set. Describe what is happening to the average clutch size as the lay date increases. When does the maximum clutch size occur for the quadratic model and what is the maximum clutch size? Discuss the reasonableness of your answers. Using the quadratic equation, predict the average clutch size for a bluebird that laid her eggs by May 28 and for a bluebird that laid her eggs by April 2. Discuss the reasonableness of your predictions.

Lay Date Average Clutch Size 60 4.4 78 4.5 96 4.6 114 4.6 132 4.5 150 4.4 166 4.0 180 3.8

Table 4.6: Average Clutch Size by Lay Date at Latitude 30

10. As indicated in problem 9, the Cornell Nest Box Network began a study in 1997 to study how clutch size (the number of eggs females lay in one nest attempt) varied by geographic location and by year for Eastern Bluebirds. They were interested in studying if clutch size increased from south to north and how the clutch sizes varied from year to year. The data points in Table 4.7 were estimated from the article Clutch-size Variation in Eastern Bluebirds (see Appendix B). Table 4.7 gives the average clutch size of the bluebirds who had laid their eggs by the indicated lay date (the lay date expresses date as days since January 1) at the latitudes of 45◦ N. Using technology produce a scatterplots for the data in Table 4.7 with the lay date as the independent 90 CHAPTER 4. QUADRATICS

variable and the average clutch size as the dependent variable. Fit a line and a quadratic to the scatterplot and discuss which equation bests ﬁt the data set. Describe what is happening to the average clutch size as the lay date increases. When does the maximum clutch size occur for the quadratic model and what is the maximum clutch size? Discuss the reasonableness of your answers. Using both the linear and the quadratic equation, predict the average clutch size for a bluebird that laid her eggs by May 28 and for a bluebird that laid her eggs by April 2. Discuss the reasonableness of your predictions and compare the predictions that you obtained from both models.

Lay Date Average Clutch Size 110 4.8 120 4.7 135 4.6 150 4.5 165 4.4 180 4.4 192 4.3 203 4.2

Table 4.7: Average Clutch Size by Lay Date at Latitude 45

11. The Cornell Nest Box Network began a study in 1997 to study how clutch size (the number of eggs females lay in one nest attempt) varied by geographic location and by year for Eastern Bluebirds. They were interested in studying if clutch size increased from south to north and how the clutch sizes varied from year to year. The data points in Table 4.8 were estimated from the article Clutch-size Variation in Eastern Bluebirds (see Appendix B).

Lay Date Average Clutch Size 60 4.7 77 4.8 94 4.85 111 4.8 128 4.75 145 4.65 162 4.45 175 4.2

Table 4.8: Average Clutch Size by Lay Date in 1997 4.6. HOMEWORK PROJECTS: 91

Table 4.8 gives the average clutch size of the bluebirds who had laid their eggs by the indicated lay date for 1997. Using technology produce a scatterplot for the data in Table 4.8 with the lay date as the independent variable and the average clutch size as the dependent variable. Fit a line and a quadratic to the scatterplot and discuss which equation best ﬁts the data set. Describe what is happening to the average clutch size as the lay date increases. What is the maximum clutch size and when does it occur? Using the quadratic equation, predict the average clutch size in 1997 for a bluebird that laid her eggs by May 28 and for a bluebird that laid her eggs by April 2. Discuss the reasonableness of your predictions. 12. As indicated in problem 11, the Cornell Nest Box Network began a study in 1997 to study how clutch size (the number of eggs females lay in one nest attempt) varied by geographic location and by year for Eastern Bluebirds. They were interested in studying if clutch size increased from south to north and how the clutch sizes varied from year to year. The data points in Table 4.9 were estimated from the article Clutch-size Variation in Eastern Bluebirds (see Appendix B). Table 4.9 gives the average clutch size of the bluebirds who had laid their eggs by the indicated lay date for 1997 and for 1998. Using technology produce a scatterplot for the data in Table 4.9 with the lay date as the independent variable and the average clutch size as the dependent variable. Fit a line and a quadratic to the scatterplot and discuss which equation best ﬁts the data set. Describe what is happening to the average clutch size as the lay date increases. What is the maximum clutch size and when does it occur? Using the quadratic equation, predict the average clutch size in 1998 for a bluebird that laid her eggs by May 28 and for a bluebird that laid her eggs by April 2. Discuss the reasonableness of your predictions.

Lay Date Average Clutch Size 60 4.9 77 5 94 5 111 4.85 128 4.75 145 4.6 162 4.3 175 3.95

Table 4.9: Average Clutch Size by Lay Date in 1998 92 CHAPTER 4. QUADRATICS

13. In this project we are going to study the relationship between time of day and temperature and time of day and humidity. For your home town, find the hourly temperature and humidity readings from approximately 7 a.m. until 11 p.m. for some day this month. You should be able to find the information at this website: http://www.wunderground.com/history/. For some cities, the readings may be given on the half hour or quarter hour - if so, adjust the time periods accordingly (e.g. 7:30 a.m. to 11:30 p.m. or 7:15 a.m. to 11:15 p.m.). If you are unable to find the information for your home town then use a nearby city instead. In any case, make sure to indicate which city and which day you are using. Using technology produce two scatterplots for the data - one for temperature and one for humidity - with the time as the independent variable. You will need to enter the time as the number of hours since midnight. For example, 1 p.m. should be entered as 13. Fit a line and a quadratic to each scatterplot and discuss which equation best fits the data set. De- scribe what is happening to the temperature and the humidity as the time increases during the day. For the temperature scatterplot, what is the predicted maximum temperature and when does it occur? Is your answer reasonable? Explain. For the humidity, what is the predicted minimum humidity and when does it occur? Is your answer reasonable? Explain. Using the quadratic equation, predict the temperature and humidity for 3:30 p.m. and for 3:30 a.m. Discuss the reasonableness of your predictions.

14. In a chemical system that has reached equilibrium, the ratio of the concentrations of the various substances are constant, K. The value of K is called the equilibrium constant. The value of K depends on the particular reaction, the temperature, and the units used to describe concentration. Usually, only the initial concentrations of the chemicals and the value of Kc are known. The general method of calculating equilibrium concentrations from initial concentrations and the value of the equilibrium constant is to let x equal one of the unknown equilibrium concentrations.

A chemical sample is allowed to decompose. The value of Kc for this 10 equilibrium is 2.2 10− at 100 degrees Celsius. If x represents the equilibrium constant for× Carbon Monoxide in this problem, the equilibrium constant expression gives the following quadratic equation

2 x 10 =2.2 10− 0.095 x × − a. Use the quadratic formula to solve this equation for x. Since concentrations can not be negative, the value for x must be positive.

b. Chemists simplify this problem by noting that because the value of Kc is very small, not much of the original solution is decomposed at equilibrium. They thus assume that x is negligible compared to 0.095 and use this 4.6. HOMEWORK PROJECTS: 93

assumption to simplify the above equation to

2 x 10 =2.2 10− 0.095 × Solve this equation for x without using the quadratic formula. c. Compare your answers to part a and b and discuss the reasonableness of using the simplified equation. 15. In this project you are going to use a pair of dice, 100 red glass beads, and 200 green glass beads to model population dynamics. You may check out the glass beads and dice from your instructor. This activity originally appeared in Investigating Evolutionary Biology in the Laboratory and was written by Brian Alters (See Appendix B). Each red glass bead represents an individual that is capable of reproducing and each green glass bead represents a resource that an individual needs to survive each year. Starting with 5 individuals (5 red beads) and 20 resource beads (20 green beads), roll the dice once for each individual. The chart below indicates what happens for that roll. After 5 rolls, the first year has ended. Take away one resource bead (green bead) for each of the individuals that were in the population at the beginning of the year (in this case 5). If there are more individuals than resource beads, those individuals die or emigrate out of the population and you should remove enough individuals so that you have the same number of individuals as resource beads. Count the number of individuals (red glass beads) left in the population. Record the number of individuals in the population at the end of the first year, the roll of the die, and the number of resource beads left at the end of the year in the table provided on the next page. You are going to continue to repeat this experiment for twenty years. For the next year add 20 resource beads to your supply. Roll the dice once for each individual in the population at the beginning of the year. After rolling the dice one time per individual and you have performed the indicated moves, the year has ended. Again, take out one resource bead for every individual that was present in the population at the beginning of the year. If there are more individuals than resource beads, those individuals die or emigrate out of the population and you should remove enough individuals so that you have the same number of individuals as resource beads. Count the number of individuals (red glass beads) left in the population. Record the number of individuals in the population at the end of the year, the roll of the die for that year, and the number of resource beads left at the end of the year in the table on the next page and repeat this procedure. Using technology make a scatterplot of the data set in the table with the year as the independent variable and the number of individuals in the population as the dependent variable. Does the data set look linear or quadratic or neither? Fit a line and a quadratic to this data set and 94 CHAPTER 4. QUADRATICS

Number on dice Move 2 Remove 5 resource beads 3 Add 1 individual for a birth 4 Add 1 individual for an immigration 5 Add 1 individual for a birth 6 Add 1 individual for a birth 7 Add 1 individual for a birth 8 Remove 1 individual for a death 9 Remove 1 individual for a death 10 Remove 1 individual for an emigration 11 Remove 1 individual for a death 12 Add 5 resource beads

discuss if either equation is a good ﬁt to the data set. Does it look like your population is going to increase indeﬁnitely or eventually die out? 4.6. HOMEWORK PROJECTS: 95

Year Numbers Rolled Number of Individuals Number of Resource Beads 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 96 CHAPTER 4. QUADRATICS Chapter 5

Exponentials

5.1 Objectives:

After completing this chapter, you should be able to

Understand positive and negative exponents • Manipulate exponential powers according to the appropriate rules • Use a variety of bases, including 10, 2, and e • Graph exponential functions • Understand the physical uses for the exponential function • 5.2 Motivation:

Exponential functions are used to model a wide range of phenomena and have been studied for many years. Sir Isaac Newton, Gottfried Wilhelm Leibniz, and Johann Bernoulli studied exponential functions in the late 1600’s and early 1700’s. In 1748, Leonard Euler introduced the letter e for the irrational number 2.718281828459... which is used as one of the bases for exponential functions. In addition to discussing various properties of this exponential function, he computed e to 23 places without the aid of a calculator or computer. Even in Euler’s time, many uses for the exponential function were well known. One of the reasons we refer to the letter e as “natural” is because it occurs in a wide variety of applications. Exponential functions in general are used in business to describe such things as medicare costs, inﬂation, and com- pound interest. In music, exponentials describe the relationship of the octave to the vibration frequency. In the sciences, exponential functions are used to describe the concentration of medicine in the bloodstream, radioactive decay, and population growth.

97 98 CHAPTER 5. EXPONENTIALS

In the next section, we will discuss how exponents and exponential functions are used to model population growth.

5.3 Introduction to Population Growth:

Exponents are used extensively in the study of population growth. A population is a group of individuals of the same species at a certain place during a specific point in time. For example, the estimated human population of the United States on May 17, 2013 at 3:39:40 was 315,655,265 (U.S. Census Bureau, 2013). Changes in population size, or population growth, may be positive or negative. Several factors influence changes in population growth including natality (number of births), mortality (number of deaths), emigration (number leaving the defined geographic area), and immigration (number moving into the defined geographic area). For example, you can go to the U.S. Census Bureau’s home page on the Internet (www.census.gov) and access the current estimated size of the U.S. population; however, if you go back to the Internet several minutes later, the number will have changed. For the date mentioned above, the U.S. Census Bureau estimated that a baby was born every 8 seconds, someone died every 12 seconds, and a new international immigrant entered the country every 44 seconds (net movement because people also are emigrating out of the country) resulting in a net gain of one person every 14 seconds. Therefore, whenever the term population is used, it is important to know the geographic and temporal component to which it pertains. We need to know something about populations and population growth because they have a direct impact on our lives. Table 5.1 provides the population size of the United States at 10-year intervals from 1900-2010. All estimates are rounded to the nearest 1,000 individuals. One reason citizens should be concerned about changes in the human population in each state is because it impacts the number of representatives that each state has in the U.S. Congress. For example, based on the 2010 Census, New York lost two of its congressional seats in the U.S. House of Representatives because the human population grew faster in states like Texas, Florida, Georgia, and Arizona than in New York. Population changes also impact the proportion of federal money allocated based on population size, planning for school districts, estimating tax revenues, and estimating expenditures that must be funded.

5.4 Demonstration:

In this demonstration we will try to determine what type of equation best describes population growth. The two types of functions that we have studied so far are lines and quadratics. Is population growth modeled by either of these types of functions? To answer this question, we are going to compare what happens when we use a line and a quadratic function to model the population growth in the United States. To make the numbers more manageable, let x =0 5.4. DEMONSTRATION: 99

Year United States Year United States 1790 3,900,000 1920 106,466,000 1800 5,300,000 1930 123,077,000 1810 7,200,000 1940 131,954,000 1820 9,600,000 1950 151,868,000 1830 12,800,000 1960 179,975,000 1840 17,000,000 1970 206,827,000 1850 23,200,000 1980 229,466,000 1860 31,400,000 1990 248,710,000 1870 38,600,000 2000 281,422,000 1900 76,094,000 2010 308,746,000 1910 92,407,000

Table 5.1: Estimated size of the human population in the United States for the ﬁrst year in the table (1790). You need to rewrite the other years in terms of how many years they are after 1790. For example, the year 1800 will be entered as x = 10.

Exploration Question #1: Using technology, make a scatterplot with the number of years since 1790 as the independent variable and with the population as the dependent variable. Fit a line to this data. What is the R2 value for the line? What is the adjusted R2 value for the line? What is the pattern of the slopes between each consecutive pairs of points? How does this tell you that the data is not linear?

Exploration Question #2: Using technology, ﬁt a quadratic function to this data set. What is the R2 value for the quadratic? What is the adjusted R2 value for the quadratic? (It may not be possible to answer this question if you are using a graphing calculator.) Use this equation to predict the United States population in 1700. Does this prediction for 1700 seem reasonable? Why does the quadratic not predict the population in 1700 with any degree of accuracy? Use the equation to predict the United States population a year from today. Does this prediction seem reasonable?

As you can see from Exploration Question #2, a parabola does not always accurately describe population growth. An equation that does a better job modeling population growth, along with describing many other situations, is an exponential equation. 100 CHAPTER 5. EXPONENTIALS 5.5 The Mathematics: 5.5.1 Integer Exponents The ﬁrst thing we need to understand is what a positive exponent represents. An exponent is shorthand notation for counting how many things you want to multiply. For example 102 is shorthand for saying you want to multiply two 10’s together, as in 10 10 = 100. In this example 10 is called the base and 2 is called the exponent. For· another example, 25 is 2 2 2 2 2 = 32. The most common numbers used for the base are 10, 2 and e· (You· · will· need a calculator or computer to work with the number e.) Exercises : Evaluate the following:

1. 34 = 5. e3 = 2. ( 4)3 = (use your calculator on 5) 3. 10−4 = 6. ( 2)8 = 1 5 − 2 4. ( 2 ) = 7. (.4) = There are “rules” of exponents that arise because of the nature of multiplication. For example, suppose you want to evaluate 23 25. If you write this out in terms of multiplication, what you have is ·

23 25 =2 2 2 2 2 2 2 2 · · · · · · · · 3 5 | {z } | {z } or eight 2’s multiplied together. The 8 comes from adding 3 and 5. The way to write this mathematically would be 23 25 =28. The fact that the base used in this example is 2 wasn’t important, but· it is important that the bases of each part are the same. What if you wanted to evaluate 23 54? If you write this out in terms of multiplication, what you have is ·

23 54 =2 2 2 5 5 5 5 · · · · · · · and there is no shorter way to write this using exponents. This leads us to our ﬁrst “rule of exponents.”

Exponential Rule 1: When multiplying like bases, you add exponents.

or, written mathematically,

Exponential Rule 1: an am = an+m. · 5.5. THE MATHEMATICS: 101

The ﬁrst rule deals with multiplying like bases. What if we need to evaluate an expression like (23)4? Writing this out, we have

(23)4 =23 23 23 23 =2 ·2 2· 2 ·2 2 2 2 2 2 2 2 . · · · · · · · · · · · 3 3 3 3 | {z } | {z } | {z } | {z } Now there are twelve 2’s, which comes from 3 times 4. So here is the second “rule of exponents”.

Exponential Rule 2: When raising a base to more than one power, you multiply exponents.

or, written mathematically,

Exponential Rule 2: (an)m = anm.

We have one more “rule” involving multiplication. What happens if a product is raised to an exponent? For example, what if we need to evaluate (2 5)3. Again, we write this out to get ·

(2 5)3 = (2 5) (2 5) (2 5). · · · · · · If we rearrange the order of the multiplication (which we can do since multiplication is commutative) we now get

(2 5) (2 5) (2 5) = (2 2 2) (5 5 5)=23 53. · · · · · · · · · · · We say that the exponent distributes to each of the terms. The same thing happens if a fraction is raised to an exponent. For example, 2 3 5 would be written out as 2 2 2 . 5 · 5 · 5 Combining the twos in the numerator and the ﬁves in the denominator, we could rewrite this as 23 . 53 Again, the exponent distributes to the numerator and the denominator.

m m Exponential Rule 3: (a b)m = am bm and a = a (for · · b bm b = 0). 6 102 CHAPTER 5. EXPONENTIALS

Exercises : Simplify the following using the ﬁrst three exponential rules.

8. x3x5 = 9. y4y7 = 10. (x3)4 = 11. (xy)5 =

3 2 t 12. a5 = 13. (x2)5 x4 = · 14. (x2y)3(xy)2 = 15. x2x4x5 =

Our next “rule” involves division. Instead of multiplying things with like bases, suppose you need to divide. For example, suppose you need to evaluate 7 2 23 . Again, if we write out what this stands for we have

7 27 2 2 2 2 2 2 2 = · · · · · · . 23 z 2 }|2 2 { · · 3 | {z } If we cancel three 2’s from the top and the bottom of this fraction we have 2 2 2 2 2 2 2 · · · · 6 · 6 · 6 =2 2 2 2=24 2 2 2 · · · 6 · 6 · 6 7 2 4 so 23 =2 . Here the 4 comes from 7 3. Again, the fact that in this example the base was 2 is not important, just− that the base of the denominator and the numerator were the same.

Exponential Rule 4: When dividing like bases, you subtract exponents.

or, written mathematically,

an n m Exponential Rule 4: am = a − . The last thing to notice about this rule is that in the subtraction the order is important. You take the exponent of the numerator and subtract from it the exponent of the denominator. This will be important for Rule 6. The next two “rules” aren’t really rules at all, but are needed to make sure 4 2 the previous rule always makes sense. For example, what if we evaluated 24 by 5.5. THE MATHEMATICS: 103

4 2 4 4 0 Exponential Rule 4? Subtracting exponents we would get 24 =2 − =2 , so it is important to ﬁgure out what 20 should mean. If instead of using Rule 4, we wrote everything out we would get

24 2 2 2 2 16 = · · · = =1. 24 2 2 2 2 16 · · · So, it turns out that 20 had better be 1, or our previous rule is nonsense.

Exponential Rule 5: Anything nonzero raised to the zero power is 1.

or, written mathematically,

Exponential Rule 5: a0 = 1 (a = 0). 6 The last “rule” is again a convention to make sure that Rule 4 always makes sense. Now what happens if we have more 2’s in the denominator of the fraction? 4 2 For example, what if we need to evaluate 27 ? Using Exponential Rule 4, we 4 2 4 7 3 would get 27 =2 − =2− . The negative in the exponent might tempt you to try subtracting in the other order, but remember it is always the exponent of the numerator minus the exponent of the denominator. So, what does it mean to have a negative in the exponent? Writing out the expression gives us

24 2 2 2 2 = · · · . 27 2 2 2 2 2 2 2 · · · · · · Instead of multiplying out the 2’s, let’s cancel the four 2’s instead. That gives us 2 2 2 2 1 1 6 · 6 · 6 · 6 = = . 2 2 2 2 2 2 2 2 2 2 23 6 · 6 · 6 · 6 · · · · · Comparing this with the answer we got from Rule 4, we see that the -3 in the exponent indicates that the three 2’s should be in the denominator. This gives us our last “rule” for exponents. This is one rule where it is easier to state the concept mathematically than it is to state it in words.

Exponential Rule 6: Any nonzero base raised to a negative exponent is equivalent to the reciprocal of that same base raised to the corresponding positive exponent.

or, written mathematically,

n 1 Exponential Rule 6: a− = n (a = 0). a 6 104 CHAPTER 5. EXPONENTIALS

Exercises : Use the rules of exponents to simplify the following. Leave your answer in exponential notation where all the exponents are positive.

14 2 16. 212 = 4 17. x− =

8 t 18. t2 = 19. 50 =

2 3x y 20. 6x4y3 =

2 3 (t ) 21. t6 =

−3 −2 x x 22. x−7 =

3 −2 4 (z ) z 23. z =

5 4 24. 210 =

2 (xy) 25. y2 = 5.5. THE MATHEMATICS: 105

5.5.2 Graphs In addition to simplifying expressions involving exponents, we will also need to be able to graph equations of the form y = bx where b is a positive number. These graphs will basically have the same shape which is either

The top graph is the graph of y = bx when b is greater than 1. The lower graph is the graph of y = bx when b is less than 1. Since every value of x determines a unique value of y in the equation y = bx, this equation is a function. In functional notation we would write this equation as f(x) = bx. 106 CHAPTER 5. EXPONENTIALS

Exercises :

26. Evaluate f(x)=2x for each of the following values of x. Use the resulting ordered pairs to plot the graph of f(x)=2x.

x f(x)=2x -4 -2 -1 0 1 2 4 8

1 x 27. Find some values for f(x) = 2 to ﬁll in the following chart. Then use 1 x these values to plot the graph off(x) = 2 . This time you must choose the x values. Make sure to choose both positive and negative values for x. −

1 x x f(x) = 2

28. Compare the two graphs. What similarities do you see? What are the diﬀerences?

29. Graph y =3x on graph paper. 30. Graph f(x)=(0.8)x on graph paper.

31. Explain why y =1x is not considered an exponential function. 5.6. HOMEWORK PROJECTS: 107 5.6 Homework Projects:

For each assigned project, write your answer in complete sentences. Pro-jects will be graded based on thoroughness and neatness as well as accuracy. Make sure to show your work in any calculations that you do.

1. Population growth also can be affected by social and political events. Ta- ble 5.2 gives the population for Oklahoma at 10-year intervals starting in 1900. To make the numbers more manageable, let x = 0 for the year 1900, and using technology, make a scatterplot with the number of years since 1900 as the independent variable and the population as the dependent variable. Using technology, fit a line, a quadratic, a cubic, and an exponential to this data set. Use each of these equations to predict the population a year from today. Which equation fits this data set best? Explain the social and political forces that would make this data set not exponential.

Year Oklahoma Population 1900 800,000 1910 1,671,000 1920 2,055,000 1930 2,401,000 1940 2,325,000 1950 2,229,000 1960 2,336,000 1970 2,619,000 1980 3,096,000 1990 3,147,000 2000 3,451,000 2010 3,751,000

Table 5.2: Estimated size of the human population in Oklahoma

rt 2. Exponential growth is given by the formula P = P0e , where P0 is the initial population, r is the rate of growth as a decimal, t is the time, and P is the population after t units of time. It can be approximated by P = P0(1 + rt). If the initial population is 100, and the rate of growth is 8%, graph both of the these curves on the same graph for t = 0 to t = 15. For what values of t does the linear function do a good job of approximating the actual exponential growth? A quadratic approximation 1 2 2 for exponential growth is P = P0(1+rt+ 2 r t ). Now graph the quadratic and the exponential curve on the same graph for t =0 to t = 15. For what values of t does the quadratic function do a good job of approximating the actual exponential growth? 108 CHAPTER 5. EXPONENTIALS

3. In this project we are going to compare the difference between using a quadratic function and an exponential function to predict the speed of a computer’s CPU (Central Processing Unit). Table 5.3 gives the name of the CPU, the date it was first issued, and how many Millions of Instruc- tions Per Second (MIPS) that the CPU could carry out. Let x = 0 for the year 1971 and using technology, make a scatterplot with the number of years since 1971 as the independent variable and let the MIPS be the dependent variable. Using technology, fit a quadratic equation to this data set. Using this quadratic equation, predict the MIPS for a computer released a year from today and for a computer released in the year 2052. Using technology, fit an exponential to this data set. Use this equation to predict the MIPS for a computer released a year from today and for a computer released in the year 2052. Compare your quadratic predictions with the exponential predictions. Which equation do you think gives the better predictions, the quadratic or exponential? Explain.

CPU Date MIPS 4004 1971 0.06 8008 1972 0.06 8080 1974 0.64 8086 1978 0.75 80286 1982 2.66 Intel 386DX 1985 5 Intel 486DX 1989 20 Intel DX2 1992 54 Pentium 1993 112 Pentium Pro 1995 200 Pentium II 1997 300 Pentium III 1999 500 Pentium 4 2000 1,700 Pentium 4/2400 2001 2,700

Table 5.3: Evolution of Processor Speed

4. In this project we are going to compare the difference between using a quadratic function and an exponential function to model the growth in the number of utility patents issued in the United States. Table 5.4 gives the first U.S. utility patent number by year. For more information on patents, see http://www.uspto.gov/web/offices/ac/ido/oeip/taf/issuyear.htm. Let x = 0 for the year 1856 and using technology, make a scatterplot with the year as the independent variable and the first patent number issued as the dependent variable. Using technology, fit a quadratic equation to this data set. Using this quadratic equation, predict the number of the first 5.6. HOMEWORK PROJECTS: 109

utility patent that will be issued a year from today. Using the quadratic equation, predict the number of the ﬁrst utility patent issued in 1836 and predict the number of the ﬁrst utility patent issued in 1960.

Year Patent Number Year Patent Number 1856 14,009 1916 1,166,419 1857 16,324 1926 1,568,040 1858 19,010 1936 2,026,516 1859 22,477 1946 2,391,856 1860 26,642 1956 2,728,913 1861 31,005 1966 3,226,729 1866 51,784 1976 3,930,271 1876 171,641 1986 4,562,596 1886 333,494 1996 5,479,658 1896 552,502 2006 6,981,282 1906 808,618

Table 5.4: First U.S. utility patent number in each calendar year

Using technology, fit an exponential to this data set and use this equation to predict the number of the first utility patent that will be issued a year from today. Using the exponential equation, predict the number of the first utility patent that was issued in 1836 (the first utility patent was actually issued in 1836) and predict the number of the first utility patent that was issued in 1960. Compare your quadratic predictions with your exponential predictions and the actual utility patent numbers that have been issued. Which equation do you think gives the better predictions, the quadratic or exponential? Explain.

5. In this project we are going to compare the difference between using a quadratic function and an exponential function to predict the number of transistors in a computer. Table 5.5 gives the name of the CPU, the date it was first issued, and the number of transistors (in thousands) in that computer. Let x = 0 for the year 1971 and using a technology, make a scatterplot with the number of years since 1971 being the independent variable and the number of transistors in a computer x years after 1971 for the dependent variable. Using technology, fit a quadratic equation to this data set. Using this quadratic equation, predict the number of transistors in a computer released a year from today and for a computer released in the year 2052. Using technology, fit an exponential to this data set. Use this equation to predict the number of transistors in a computer released a year from today and for a computer released in the year 2052. 110 CHAPTER 5. EXPONENTIALS

CPU Date Transistors (in thousands) 4004 1971 2.3 8008 1972 3.5 8080 1974 4.5 8086 1978 29 80286 1982 134 Intel 80386 1985 275 Intel 80486 1989 1,200 Intel DX2 1992 1,200 Pentium 1993 3,100 Pentium Pro 1995 5,500 Pentium II 1997 7,500 Pentium III 1999 9,500 Pentium 4 2000 42,000 Itanium 2 McKinley 2002 220,000 Core 2 Duo 2006 291,000 Core i7 (Quad) 2008 731,000 Quad-Core + GPU Core i7 2011 1,160,000 Quad-Core + GPU Core i7 2012 1,400,000

Table 5.5: CPU vs. Number of transistors

Compare your quadratic predictions with your exponential predictions. Which equation do you think gives the better predictions, the quadratic or exponential? Explain. 6. In this project we are going to compare the difference between using a quadratic function and an exponential function to predict the blood pressure of an animal relative to the average height of its head above its heart. Table 5.6 gives the average blood pressure of various animals (measured in mm Hg) and the average height of its head above its heart (measured in mm). Using technology, make a scatterplot with the height of the head above the heart as the independent variable and the blood pressure as the dependent variable. Using technology, fit a quadratic equation to this data set. Using this quadratic equation, predict the blood pressure for a dog which has a height of 200 mm for its head above its heart and for a giraffe which has a height of 3000 mm for its head above its heart. Using technology, fit an exponential to this data set. Use this equation to predict the blood pressure for a dog which has a height of 200 mm for its head above its heart and for a giraffe which has a height of 3000 mm for its head above its heart. Compare your quadratic predictions with your exponential predictions. Which equation do you think gives the better prediction, the quadratic or 5.6. HOMEWORK PROJECTS: 111

Animal Blood Pressure Height of Head Above Heart Human 120 500 Cow 157 500 Duck 162 100 Cat 129 100 Guinea Pig 60 25 Goat 98 400 Monkey 140 200 Turkey 193 300

Table 5.6: Blood Pressure, Selected Animals

exponential? Explain.

7. In this project we are going to compare the diﬀerence between using a quadratic function and an exponential function to predict the average heart rate of an animal relative to its weight. Table 5.7 gives the average heart rate of various animals (measured in beats per minute) and its average weight (measured in grams).

Animal Average Heart Rate Weight Human 60 90,000 Cow 65 800,000 Hamster 450 60 Cat 150 2,000 Chicken 275 1,500 Horse 44 1,200,000 Rabbit 205 1,000 Elephant 30 5,000,000

Table 5.7: Average Heart Rate, Selected Animals

Using technology, make a scatterplot with the weight as the independent variable and the average heart rate as the dependent variable. Using technology, fit a quadratic equation to this data set. Using this quadratic equation, predict the average heart rate for a small dog which has a weight of 2,000 grams and for a giraffe which has a weight of 900,000 grams. Using technology, fit an exponential to this data set. Use this equation to predict the average heart rate for a small dog which has a weight of 2,000 grams and for a giraffe which has a weight of 900,000 grams. Compare your quadratic predictions with your exponential predictions. Which equation do you think gives the better prediction, the quadratic or exponential? Explain. 112 CHAPTER 5. EXPONENTIALS

8. Newton’s law of cooling describes the cooling rate of a body relative to the temperatures of the object and the surroundings. The following data in Table 5.8 was collected for a container of water at an initial temperature of 100◦C placed inside a freezer compartment at 0◦C. The time is measured in minutes and the temperature is measured in degrees Celsius.

Time Temp Time Temp Time Temp 0 100 40 67 80 45 2 98 42 66 82 44 4 96 44 64 84 43 6 94.5 46 63 86 42 8 92.5 48 62 88 41.5 10 90.5 50 61 90 40.5 12 88 52 59 92 40 14 87 54 58 94 39 16 85 56 57 96 38.5 18 83.5 58 56 98 37.5 20 82 60 55 100 36.5 22 80 62 53.5 102 36 24 78.5 64 53 104 35.5 26 77 66 52 106 34.5 28 75 68 50.5 108 34 30 74 70 49.5 110 33 32 73 72 48.5 112 32.5 34 71 74 47.5 114 32

Table 5.8: Newton’s Law of Cooling

Using technology, make a scatterplot with time as the independent variable and temperature as the dependent variable. Using technology, ﬁt a quadratic equation to this data. Use this equation to predict the water temperature after 2 hours, 3 hours, and 4 hours. Using technology, ﬁt an exponential to this data set. Use this equation to predict the water temperature after 2 hours, 3 hours, and 4 hours. Compare your quadratic predictions with your exponential predictions. Which equation do you think gives the better prediction, the quadratic or exponential? Explain.

9. In this project we are going to compare the diﬀerence between using a quadratic function and an exponential function to extrapolate population size. Table 5.9 gives the population of whooping cranes observed during the winter count at the Aransas National Wildlife Refuge in Texas from 1940-2012. Let x = 0 for the year 1940 and using technology make a scatterplot with 5.6. HOMEWORK PROJECTS: 113

the number of years since 1940 as the independent variable and the number of whooping cranes observed for that year as the dependent variable. Using technology ﬁt a quadratic equation to this data set. Using this quadratic equation, predict the number of whooping cranes a year from today and for the year 1900. Using technology, ﬁt an exponential to this data set. Use this equation to predict the number of whooping cranes a year from today and for the year 1900. Compare your quadratic predictions with your exponential predictions. Which equation do you think gives the better prediction, the quadratic or exponential? Explain.

Year Number Year Number 1940 22 1999 185 1950 34 2000 177 1960 33 2001 174 1970 56 2002 185 1980 76 2003 194 1990 146 2004 216 1991 132 2005 218 1992 136 2006 237 1993 143 2007 266 1994 133 2008 270 1995 158 2009 263 1996 160 2010 281 1997 182 2012 279 1998 183

Table 5.9: Numbers of Whooping Cranes at Aransas NWR

10. In this project we are going to compare the diﬀerence between using a quadratic function and an exponential function to extrapolate population size. In 1889 the Smithsonian estimated that there were just over 1000 bison left in the U.S. They also estimated that there had once been over 60 million bison (now we think that it was only 30 million). In 1907 ﬁfteen bison from the New York zoo were introduced into the Wichita Mountains National Forest and Game Preserve (now called the Wichita Mountains Wildlife Refuge) which consisted of 8000 acres. The preserve had been established by Teddy Roosevelt who had hunted there with Quanah Parker (a famous Comanche chief) and is located outside of Lawton, OK. 114 CHAPTER 5. EXPONENTIALS

By 1918 there were 100, in 1938 there were 330, and in 1957 there were 1000 bison. The herd continued to grow until 1970 when the staff at the Wichita Mountains Wildlife Refuge decided there were too many bison for only 8000 acres and they cut the herd back to about 700. Table 5.10 gives the population of the bison as discussed above. Let x = 0 for the year 1907 and using technology make a scatterplot with the number of years since 1907 as the independent variable and the number bison as the dependent variable. Using technology fit a quadratic equation to this data set. Using this quadratic equation, predict the number of bison a year from today and for the year 1970 when the staff decided there were too many bison. Using technology, fit an exponential to this data set. Use this equation to predict the number of bison a year from today and for the year 1970 when the staff decided there were too many bison. Compare your quadratic predictions with your exponential predictions. Which equation do you think gives the better prediction, the quadratic or exponential? Explain.

Year Number 1907 15 1918 100 1938 330 1957 1000

Table 5.10: Numbers of Bison at Wichita Mountains National Forest Chapter 6

Logarithms

6.1 Objectives:

After completing this chapter, you should be able to

Understand the deﬁnition of logarithm • Understand the relationship between logarithms and exponents • Use properties of logarithms • Graph a logarithmic function • Understand the physical uses for the logarithmic function • 6.2 Motivation:

Logarithms were discovered by John Napier, a Scotsman who lived from 1550 to 1617. Napier was an extremely colorful character, born when his father was just 16. Napier was intensely involved in the religious and political movements of his time. To relax from such activities, Napier studied mathematics and science. Logarithms were invented by Napier as a computing device to aid astronomers. In the 1500’s, all calculations were done by hand, as there were no calculators or computers, and astronomers were dealing with larger and larger numbers that made computations increasingly tedious. As we will see, logarithms deal with the exponents of numbers and, thus, reduced problems with incredibly large numbers to problems of a much more manageable size. Although we do now have calculators and computers to aid with calculations, logarithms are still found in a variety of applications. In addition to being a useful algebraic tool in dealing with exponential functions, logarithms are used in exponential decay, radioactive decay (such as carbon dating), thermodynamics, the measure of acidity in solutions (pH), Newton’s law of cooling, the Richter

115 116 CHAPTER 6. LOGARITHMS scale (which measures the magnitudes of earthquakes), and in decibels (which measure the intensity level of sound.) In the next section, we are going to use a sound level meter to determine various intensity levels of sound. We will then see why logarithms describe the intensity level of sound.

6.3 Introduction to Sound:

A sound wave in air consists of alternating regions of high pressure and low pressure (also called condensations and rarefactions) generated by vibration. The source of vibration could be vocal cords, the diaphragm of a loudspeaker, or the prongs of a tuning fork. The sound wave travels through the air but the air as a whole does not move, the molecules of air merely oscillate back and forth. The motion of the air is much the same as vibrations of a spring. The endpoints don’t move but the coils of the spring do vibrate.

This type of motion is called longitudinal. What causes the sound wave to move or propagate? The regions of high pressure and low pressure attempt to equalize with the surrounding air. The high pressure regions drop in pressure and the low pressure regions increase in pressure. However, just like the spring example, the pressure overshoots the equilibrium pressure so the wave pulls itself forward through space. In an extremely intense sound wave, such as a jack hammer or jet engine, the pressure variation from low to high is about one percent of the equilibrium pressure. The ear performs the task of converting the mechanical vibrations of a sound wave into electric nerve impulses that our brains perceive as sound. A microphone works in much the same way but so far the ear is unmatched in its ability to accommodate a wide range of intensities of sound. The ratio of the most intense sound that we can hear to the intensity of the faintest whisper is 1012, a factor of a trillion! Intensity is deﬁned as the amount of energy crossing a region of unit area 6.3. INTRODUCTION TO SOUND: 117

(usually measured in square meters) each second. The intensity can be varied several ways and the energy output by the source can be changed. One way is to turn up the volume. Another way is to change the area over which the sound is recorded, i.e. we can get a bigger ear. The third way is to change the distance to the source. For example, just as when a pebble is thrown in the pond and ripples emanate with decreasing amplitude as the radius of the ripples increase, so does the intensity of the sound diminish as the distance from the source is increased. The metric unit to describe intensity is the Watt/meter2 (W/m2). The faintest sound that the typical young adult can hear has an intensity of 12 2 10− W/m , this is called the threshold of hearing. The loudest sound that is audible for most adults is 1W/m2, this is called the threshold of pain. Intensities greater than the threshold of pain are felt rather than heard. Severe ear damage can result from long term exposure to very intense sounds. At an intensity of 10,000 W/m2 most eardrums will rupture. Mathematically, we describe intensity (I) in terms of logarithms to reduce the relative sizes of the numbers involved. Instead of comparing numbers from 12 4 0.000000000001 (10− ) up to 10,000 (10 ), we use a logarithmic scale to convert the same numbers to a range of 0 to 160. This logarithmic scale is called the intensity level or the decibel level (dB), which is deﬁned to be 12 dB = 10 log(I/10− ). As a sound wave spreads out from its source, its intensity falls oﬀ because the area of the wave front grows larger, and therefore the wave energy per unit area decreases. 118 CHAPTER 6. LOGARITHMS

As a wave front grows from r1 to r2 the area of the wave front increases 2 2 from 4πr1 to 4πr2. The total power (energy per time) carried by the wave has remained the same but the power per unit area, or the intensity, has decreased. P P Mathematically, we have that I1 = 2 and I2 = 2 . Combining these two 4πr1 4πr2 2 I2 r1 equations, we have that = 2 . This means that if a sound has an intensity I1 r2 of 0.01 W/m2 at a distance of 10 meters, the intensity will fall to 0.0025 W/m2 at distance of 20 meters. Another way to think of this concept is to look at P 1 the equation I = 4πr2 . If the distance r is doubled, the intensity I is 4 of the original value. This type of relationship occurs frequently; it is called an inverse 1 square law since, for example, 4 is the inverse of the square of 2.

6.4 Demonstration:

In class, the instructor will have a sound level meter and 5 identical doorbells. The sound level meter measures (in decibels) the intensity level output (how loud the doorbells sound.)

Exploration Question #1: Look up decibels in the diction-ary. Write down the deﬁnition. Do you understand the deﬁnition? Write in your own words what you think it means.

The instructor is going to ring ﬁrst one doorbell, then two doorbells, then three doorbells, etc., until all ﬁve doorbells are ringing. After each doorbell is added, we will record the intensity level of sound from the sound level meter.

Exploration Question #2: Compared to the reading on the sound level meter of the ﬁrst doorbell, what do you think will happen to the intensity or the loudness of the sound when there are two doorbells ringing?

Fill in the following table with the number of doorbells and the decibel reading on the sound level meter.

number of doorbells decibels

Plot these data points on a graph with the number of doorbells on the x axis and the decibels on the y axis. − − 6.5. THE MATHEMATICS: 119

Exploration Question #3: Describe the graph. Is this a graph of any of the functions that we have studied previously? Was the decibel reading for two doorbells twice as large as the decibel reading for one doorbell? How does the decibel reading for ﬁve doorbells compare to the decibel reading for one doorbell?

6.5 The Mathematics: 6.5.1 Logarithms A logarithm is another way of dealing with exponents. As we discussed in the introduction, logarithms convert large numbers into numbers that have a range that is easier to deal with. In the previous chapter, you were given a base and an exponent and were expected to ﬁnd the result. For logarithms (which we sometimes abbreviate as logs), you are given the base and the result, and you are trying to ﬁnd the exponent. As an example, we know that 23 =8. Here the base is 2, the exponent is 3, and the result is 8. The way we would write this statement using logarithms would be log2 8 = 3 (and we would say “the log base 2 of 8 is 3”). Notice where the corresponding parts go between these two statements:

23 =8

log2 8=3. Given one of these two statements, you should be able to convert to the other statement. In general we would write this as

n b = r is equivalent to logb r = n.

The important thing to remember is that when you are dealing with logs you are really dealing with exponents! 120 CHAPTER 6. LOGARITHMS

Exercises : Rewrite the following exponential statements as their corresponding logarithmic statements.

1. 34 =81 5. a0 =1 2. 41/2 =2 6. x4 = y 3. 104 =10000 7. 272/3 =9 4. 51 =5 8. bx = y

Rewrite the following logarithmic statements as their corresponding exponential statements.

9. log2 8=3 14. log6 36=2 1 10. log3 1=0 15. log2 8 = 3 1 1 − 11. log9 3 = 2 16. log644 = 3 12. log .1 = 1 17. log x = y 10 − b 13. log4 y = a 18. loga 1=0

Evaluate the following logarithms.

19. log2 16= 22. log2 1 = 1 20. log3 3= 23. log5 25 = 1 21. log3 81= 24. log9 9 =

Just as we had “rules” for exponents, we have “rules” for logarithms. These aren’t actually new rules at all; they are just the restatement of the exponential rules. Remember that the ﬁrst exponential rule was when multiplying like bases, add exponents. So when we add logarithms with like bases, we end up multiplying the results. The revised Exponential Rule 1 becomes

Logarithm Rule 1: logb r +logb s = logb(r s) (where b > 0,b =1, and r, s > 0). · 6

Compare this with the example log2 8+log2 128 =? If we just convert each 3 log statement, we know that log2 8 = 3 since 2 = 8 and that log2 128 = 7 7 since 2 = 128. So log2 8 + log2 128 = 3+7 = 10. If we had used Logarithm Rule 1 on log2 8 +log2 128 to ﬁrst combine this into one logarithm, we would have log2 8 +log2 128 = log2(8 128) = log2 1024. So, what is log2 1024? Since 10 · 2 = 1024, we know that log2 1024 = 10 (which you certainly should expect!) What about Exponential Rule 4? This rule stated that when dividing like bases, you subtract exponents. So when we subtract logarithms with like bases, we end up dividing the results. Just as in the exponential rule, order here is important. The revised Exponential Rule 4 becomes 6.5. THE MATHEMATICS: 121

r Logarithm Rule 2: logb r logb s = logb s (where b > 0, b = 1, and r, s > 0). − 6

Compare this with the example log2 1024 log2 128 =? If we just convert − 10 each log statement, we know that log2 1024 = 10 since 2 = 1024 and that log 128 = 7 since 27 = 128. So log 1024 log 128 = 10 7=3. If we had 2 2 − 2 − used the logarithm rule 2 on log2 1024 log2 128 to ﬁrst combine this into one − 1024 logarithm, we would have log2 1024 log2 128 = log2 128 = log2 8=3. Either way, you come up with the same answer!− The revised Exponential Rule 5 becomes

Logarithm Rule 3: logb 1=0(b > 0).

Our fourth logarithm rule is a little diﬀerent from the previous three. Instead of coming from one of the exponential rules, it actually comes from Logarithm 4 Rule 1. Let’s start by looking at an example: log3(9 ). If we use what we know about exponents and logs we can rewrite this as

log (94) =log (9 9 9 9). 3 3 · · · Using Logarithm Rule 1, we can convert the multiplication into the addition of logarithms.

log (9 9 9 9) =log 9+log 9+log 9+log 9 3 · · · 3 3 3 3 4 . =4 log 9 | · 3 {z } Notice that the exponent that was originally on the 9 moved down in front of the logarithm. Here exponentiation is converted to multiplication. Written mathematically we have

Logarithm Rule 4: log (rs) = s log r (b > 0, b =1, r > 0). b · b 6 122 CHAPTER 6. LOGARITHMS

Exercises : Use the four rules of logarithms to simplify the following. Write your answer in the form of a single logarithm with no exponentiation.

3 25. logb 7+logb 3= 28. logb 5+logb 5 = 26. logb 14 logb 2= 29. logb a logb c + logb d = 5 − − 27. logb x = 30. logb 2+logb 4 =

6.5.2 Common Logs and Natural Logs There are two bases for logarithms that are used more frequently in the natural and physical sciences; these are base 10 and base e. A logarithm which has a base of 10 is called a common logarithm and many times the base 10 will not be written. For example, instead of writing log10 100 we could write log100 and it would be understood that the base is 10. Most calculators have a log key that is the common logarithm. The other base that appears quite often is the base of e. With this base, the logarithm is called the natural logarithm. To distinguish this logarithm from the common logarithm, the abbreviation ln is used in place of loge . For example loge 8 is written as ln8. Most calculators have a ln key as well. These logarithms work the same way as all the other logarithms that were discussed in the previous section. All of the rules of logarithms still hold.

Exercises : Evaluate the following logarithms.

31. log10= 34. ln1 = 32. log1000= 35. ln e = 33. log(.1)= 36. ln e3 =

Evaluate the following logarithms using a calculator or a computer. Round your answer to two decimal places.

37. log5= 40. ln3 = 1 38. log(.3)= 41. ln 2 = 39. log20= 42. ln9 = 6.5. THE MATHEMATICS: 123

6.5.3 Graphs

The graph of any logarithmic function of the form y = logb x will basically have the following shape as long as the base b is greater than 1.

If the base is less than 1, the graph is reﬂected about the x axis and has the following shape. −

Remember that the base b must be positive and not equal to 1. The other restriction is that the x value must be positive to get a real number for y.

Exploration Question #5: What happens if you try to have your calculator evaluate the log of a negative number or 0? Use your calculator to ﬁnd the common logarithm of 0. What does the display read? Now try to ﬁnd the common logarithm of -2. What does the display read? 124 CHAPTER 6. LOGARITHMS

Exercises :

43. Evaluate log2 x for each of the following values of x. Use the resulting ordered pairs to plot the graph of f(x) = log2 x.

x log2 x 1/8 1/4 1/2 1 2 4 8 16

44. Find the values for the following logarithm to ﬁll in the following chart. Then use these values to plot the graph of f(x) = log1/2 x.

x log1/2 x 1/8 1/4 1/2 1 2 4 8 16

45. Compare the two graphs. What similarities do you see? What are the diﬀerences? 46. Graph y = log x and y = ln x. Compare these two graphs with each other and the two graphs above. What similarities do you see? What are the diﬀerences? 6.5. THE MATHEMATICS: 125

6.5.4 Equations Logarithms are quite useful in solving exponential equations, especially when the unknown variable is in the exponent. In the chapter on exponentials, we discussed many situations in which exponential equations are found and logarithms are needed for the manipulation of these equations. Conversely, if we have an equation in which we are taking the logarithm of a variable, then exponentiation is used to solve for the variable. Let’s start with trying to solve an exponential equation where the variable is in the exponent. For example, suppose we need to ﬁnd a number for x, where 10x = 130. There is no integer that we can substitute for x so that 10 raised to that number is 130. Besides just guessing what x should be, we can use logarithms to determine the exact value of x. Since we have an equation, we can take the logarithm of both sides. Although it doesn’t matter which base for the logarithm that you use, most people either use the common log or the natural log since those are keys on their calculators. Since 10x = 130 has a base of 10, most people would take the common log of both sides. Starting with

10x = 130 we take the common log of both sides to get

log10x = log130.

Using Logarithm Law 4, we bring down the x in front of the common log on the left hand side to get x log10 = log130. We know that log10 = 1 but we won’t be able to compute log130 without either a calculator or a computer. This gives us

x(1) = log130 or x = log130 2.11394. ≈ You should use your calculator to conﬁrm that 102.11394 is approximately 130. Also notice that 2.11 is between 2 and 3 which coincides with 130 being between 102 = 100 and 103 = 1000. For another example, let’s solve 2e0.1x = 60. Again, we would like to take the log of both sides; here we would probably take the natural log of both sides since the base is e. However, we ﬁrst need to divide both sides by the 2 to isolate the exponential. Starting with

2e0.1x = 60 we divide both sides by 2 to get

e0.1x = 30. 126 CHAPTER 6. LOGARITHMS

Now that we have the exponential by itself on one side, we can take the natural logarithm of both sides which gives us

ln e0.1x = ln30.

Using Logarithm Rule 4 to bring down the exponent we have

0.1x lne = ln30.

We know that ln e = 1 and by using the calculator we get that ln 30 3.4012. Substituting these values into the equation gives us ≈

0.1x(1) 3.4012. ≈ We divide both sides by 0.1 to solve for x getting

x 34.0112. ≈ Again you should check that e raised to the (0.1)(34.0112) power is approximately 30. If you have a problem whose base is not 10 or e then you can use either the natural logarithm or the common logarithm. For example, to solve 3x = 15, we can take the natural log of both sides or the common log of both sides. If we take the common log of both sides, we get

log3x = log15.

Again, we use Logarithm Rule 4 to bring down the x in front of log3, and use the calculator to determine that log 3 0.47712 and that log15 1.17609. Substituting these into the equation we≈ have ≈

x(0.47712) 1.17609. ≈ Dividing both sides by 0.47712, we see that

x 2.4650. ≈ 6.5. THE MATHEMATICS: 127

Exercises : Solve the following equations for the unknown variable.

47. 1000 = e0.02t 50. 2x = 13 48. 103x+5 =29 51. 2000 = 1000e0.08t 0.01r 5x 49. 2e− =8 52. 10 = 320

Because of the relationship between logarithms and exponents, we can use exponentiation to help us solve problems involving logarithms. This is, in some ways, the inverse of the previous type of problem. Now we are given an equation involving a logarithm and a variable, and we need to solve for the variable. For example, suppose that we have the equation log x = 3. To solve this equation, we need to remember our definition of logarithms. Since this is the common log, the base is 10 and we have that 103 = x or that x = 1000. If we have a more complicated expression involving logarithms, the first thing that we need to do is use the rules of logarithms to rewrite the expression so that it has only one logarithm. For example, suppose we need to solve log x log3 = 8. We first need to rewrite the left hand side so that it involves only one− logarithm. Using Logarithm Rule 2 we gives us x log x log3 = log . − 3 Substituting this into our original equation, we have that x log =8. 3 Now using the definition of logarithms, we get x 108 = . 3 Multiplying both sides by 3, we see that

x = 300, 000, 000.

Exercises : Solve the following equations for the unknown variable.

53. ln x =5 56. log(2x)=1 54. ln(x +5) = 2 57. log x log3 = 1 55. ln x +ln5=− 2 58. log x −+log3=3− − 6.5.5 Rates of Growth We can use what we have learned about exponentials and logarithms to determine rates of growth (or decay) of exponential data sets. For example, let us return to the data from the United States Census Bureau regarding the population in the United States. 128 CHAPTER 6. LOGARITHMS

Table 6.1: Estimated size of the human population in the United States

rt We will use the equation P = P0e , where r is the rate as a decimal, t is time in years, P0 is the initial population, and P is the population after t units of time. To make the numbers more manageable, we can let t = 0 for the year 1790 and let P0 be the population (in millions) in 1790. Substituting P0 into rt rt P = P0e we have P = 3.9e . We know that the population in 1800 was 5.3 million and that 1800 is 10 years after 1790 (so t = 10). Substituting these two values into the above equation we get 5.3=3.9er10. We know we can solve this equation for r using logarithms. Dividing both sides by 3.9 to isolate the exponential we have 1.36 e10r. ≈ Taking the natural logarithm of both sides gives us ln(1.36) ln(e10r)=10r ln e = 10r. ≈ Dividing both sides by 10 we see that ln(1.36) r 0.03074847, ≈ 10 ≈ which is the rate of growth of the United States population from 1790 to 1800. If we substitute this value back into our original equation we have that P =3.9e0.03074847t. We can use this equation to predict the population for other years. For example if we were to use this to predict the population (in millions) for 1805 we could substitute in t = 15 to get P =3.9e0.03074847(15). 6.5. THE MATHEMATICS: 129

Simplifying this, we get that the predicted population in 1805 would be 6.1 million. ≈ Exercises :

59. Using the formula P = 3.9e0.03074847t derived from Table 6.1, predict the population of the United States in the year 1980. How close is this prediction to the actual population in 1980? 60. To see the eﬀect of rounding on the ﬁnal answer, use the formula P = 3.9e0.03t to predict the population of the United States in the year 1980. How does this prediction rounded to the nearest thousands compare with the prediction from problem 59 rounded to the nearest thousands?

rt 61. Using the formula P = P0e and Table 6.1, ﬁnd the rate of growth (r) of the United States population from the year 1960 to the year 1970. (Hint: let t = 0 correspond to the year 1960 and then t = 10 will be the year 1970.) Use this rate to predict the United States population in the year 1980. How close is this prediction to the actual population in 1980? 62. Compare the two predictions from problems 60 and 61. Which is the better prediction? Explain. What factors inﬂuence the reliability of the prediction? 130 CHAPTER 6. LOGARITHMS 6.6 Homework Projects:

1. In this project we are going to compute the rate of increase in the speed of a computer’s CPU (Central Processing Unit). Table 6.2 gives the name of the CPU, the date it was first issued, and how many Millions of Instruc- tions Per Second (MIPS) that the CPU could carry out. Assuming exponential growth, find the rate of increase from 1971 to 1974 and use this rate to predict the MIPS for a computer released a year from today and in the year 2052. Now find the rate of increase from 1997 to 2001 and use this rate to predict the MIPS for a computer released a year from today and for the year 2052. In your opinion, which calculation provides the best estimate of MIPS a year from today? Which calculation provides the best estimate of MIPS in 2052? Provide reasons that might account for differences in these calculations.

Table 6.2: Evolution of Processor Speed

2. In this project we are going to compute the rate of growth in the number of transistors in a computer. Table 6.3 gives the name of the CPU, the date it was ﬁrst issued, and the number of transistors in that computer. Assuming exponential growth, ﬁnd the rate of increase for the number of transistors from 1971 to 1972 and use this rate to predict the number of transistors for a computer released next year and in the year 2052. Now 6.6. HOMEWORK PROJECTS: 131

ﬁnd the rate of increase from 2008 to 2012 and use this rate to predict the number of transistors for a computer released next year and in the year 2052. In your opinion, which calculation provides the best estimate of number of transistors for a year from today? Which calculation provides the best estimate of number of transistors in 2052? Provide reasons that might account for diﬀerences in these calculations.

Table 6.3: CPU vs. Number of transistors

3. In this project we are going to compute the rate of growth in the number of utility patents issued in the United States. Table 6.4 gives the first United States utility patent number by year. Assuming exponential growth, find the rate of growth in the first United States utility patent number from 1859 to 1861. Using this rate, predict the first utility patent number that will be issued next year. Using this rate, find in what year the utility patent number 1,000,000 was issued. Now find the rate of growth in the first United States utility patent number from 1946 to 2006 and using this rate, predict the first utility patent number that will be issued next year. Using this rate, find in what year the utility patent number 1,000,000 was issued. In your opinion, which rate provides the best estimate of the first United States utility patent number that will be issued for next year? Which rate 132 CHAPTER 6. LOGARITHMS

provides the best estimate of the year in which the utility patent number 1,000,000 was issued? Provide reasons that might account for diﬀerences in these calculations.

Table 6.4: First U.S. utility patent number in each calendar year

4. A rancher decides to establish an exotic species hunting preserve on a 50,000-hectare ranch. The rancher purchases 50 roan antelope, a species native to Africa, and releases them on the ranch on January 1, 2005. One year later, the rancher has 70 roan antelope on the ranch. Find the rate of growth in the roan antelope from 2005 to 2006. If the population continues to grow at the same rate, what will the population size be in 2007? If this same rate of growth continues, what would be the population size in 2017? How long will it take the population of roan antelope to double? In what year will the population of roan antelope be 1000? 5. Assuming exponential growth, calculate the rate of increase for the human population in the United States from 1900 to 1910 using the data points in Table 6.1 Use this rate to estimate the population size of the United States in 2020. Using this rate, find in what year the population was 100 million. Repeat this exercise using the rate of increase from 1960 to 1970 to estimate population size in 2020 for the United States and to find in what year the population was 100 million. Which rate provided the best estimate of population size in 2020? Which rate provided the best estimate of the year when the population was 100 million? Provide reasons that might account for differences in these calculations. 6. Newton’s law of cooling describes the cooling rate of a body relative to the temperatures of the object and the surroundings. The following data in Table 6.5 was collected for a container of water at an initial temperature of 6.6. HOMEWORK PROJECTS: 133

100◦C placed inside a freezer compartment at 0◦C. The time is measured in minutes and the temperature is measured in degrees Celsius. Find the rate of decay in the water temperature. Using this rate, ﬁnd the water temperature after 2 hours. Find the water temperature after 3 hours. When was the water temperature 25◦?

Table 6.5: Newton’s Law of Cooling

7. In this project we are going to compute the rate of growth in the number of users of Facebook. Table 6.6 gives the month and the number of active users in millions at Facebook according to Facebook Inc. Assuming exponential growth, find the rate of increase for the number of active users at Facebook from December 2004 to December 2009 and use this rate to predict the number of active users at Facebook in December of 2016 and in December of 2020. Now find the rate of increase from December 2009 to December 2012 and use this rate to predict the number of active users at Facebook in December of 2016 and in December of 2020. In your opinion, which calculation provides the best estimate of number of active users at Facebook in December of 2016? Which calculation provides the best estimate of number of active users at Facebook in December of 2020? Provide reasons that might account for differences in these calculations. 134 CHAPTER 6. LOGARITHMS

Month Year Active Users Month Year Active Users December 2004 1 December 2009 350 December 2005 5.5 February 2010 400 December 2006 12 July 2010 500 April 2007 20 December 2010 608 October 2007 50 July 2011 750 August 2008 100 September 2011 800 January 2009 150 December 2011 845 February 2009 175 March 2012 901 April 2009 200 June 2012 955 July 2009 250 September 2012 1010 September 2009 300 December 2012 1060

Table 6.6: Active Users at Facebook (in millions)

8. A common circuit is composed of a resistor R, an inductor L, and a source of electromotive force . A resistor is a device used to increase the difficulty with which electricity can move through a circuit; the units of resistance are given in ohms (Ω). An inductor is a device that opposes the change in flow of electricity in a circuit; the units of inductance are given in henries (H). Electromotive force (emf) is another name for potential or voltage and has units of volts (V ). A common source of emf is a battery. Symbols for these three circuit elements are given in the figure below. The three elements are combined in what is called a series circuit.

The flow of electricity in a circuit is called current (I) and the measure of how much work a given amount of current can do in a fixed amount of time is called potential or voltage (V ). The current, IL which is a function of time, flowing through the inductor after the switch is closed is given by

tR/L IL(t) = 1 e− . R − 6.6. HOMEWORK PROJECTS: 135

The voltage, VL, as a function of time across the inductor after the switch is closed is given by tR/L VL(t) = e− . The value epsilon () is the electromotive force provided by the battery. For the questions below assume the battery provides six volts, the resistor has a value of one hundred ohms, and the inductor has a value of two hundred henries. a. Verify that the quantity tR/L is unitless. An ohm is equivalent to kg m2 kg m2 sec3 A2 and a henry is equivalent to sec2A2 , where A stands for amperes. b. Substitute the given values for , R, and L into the formula for IL(t). Then graph this equation on graph paper with t as the independent variable and IL as the dependent variable. Start your graph with t = 0 and make sure to plot enough points to get a good overall picture of the graph.

c. Substitute the given values for , R, and L into the formula for VL(t). On a separate piece of graph paper, graph this equation with t as the independent variable and VL as the dependent variable. Start your graph with t = 0 and make sure to plot enough points to get a good overall picture of the graph. d. How long before the voltage is only one volt? only 1% of the battery voltage? Compute these answers algebraically using logarithms. Then mark these times on your graph from part b. e. What is the maximum value of the current? Estimate this answer from your graph from part b. How long before the current reaches half this value? Compute this answer algebraically using logarithms. Then mark this time on your graph from part b. f. Power is a measure of how fast energy is converted from one form to another. For electrical circuits the power is the product of the current and the voltage. Algebraically ﬁnd the power in the inductor L as a function of time. On another piece of graph paper, graph this new equation with t as the independent variable and P as the dependent variable. Start your graph with t = 0 and make sure to plot enough points to get a good overall picture of the graph. g. Energy (in Joules) is the area under a power-time curve. Calculate the energy stored in the inductor by estimating the area under your graph in the previous part. 1 2 h. Compare your answer in g to the value E = 2 LIMax, where IMax is the maximum value of the current that you found in part e.

9. Astronomers and astrophysicists study objects great distances from the earth. Often the objects of study are obscured by clouds of galactic dust. Certain forms of electromagnetic radiation can penetrate this dust al- lowing scientists to “see” what is inside or behind the clouds. Gamma 136 CHAPTER 6. LOGARITHMS

radiation serves as a good probe for studying supernova remnants in our galaxy. One source of this radiation is Titanium-44 (44Ti). The amount of radiation depends directly on the amount of Titanium-44 present. An ongoing problem in physics has been the accurate measurement of how fast Titanium-44 decays. Experiments over the last 30 years have placed the half-life of 44Ti anywhere from 46 to 67 years. Two recent experiments (see Appendix B for the citation of the articles) have determined a more accurate half-life of 59.2 0.6 years. ± In this project we wish to study the eﬀects of error on exponential and logarithmic quantities. The half-life (T1/2) of a radioactive material is the measure of how long it takes for one-half of the material to decay. For example, if 10 grams of a material has a half-life of 4 minutes, then after 4 minutes, 5 g will remain. After 4 more minutes, 2.5 g will remain. Mathematically, this can be written as

1 rT1 2 P = P e− / 2 0 0

where P0 is the original amount of material, r is the decay rate, and T1/2 is the half-life. Once the rate r is determined the amount of material P (t) can be predicted for any amount of time using the equation

rt P (t) = P0e− .

a. What is the decay rate for a half-life of 46 years? What is the decay rate for a half-life of 67 years? b. What percentage of a 100 gram sample for each half-life in part a will decay in 6 months? in 1 year? in 2 years? in 5 years? c. If the mass of a radioactive sample can only be measured to within 0.5% of its actual value, what range of half-lives does this imply for a 100± gram sample which decays to a 98.7 gram sample after 1 year? (Hint: an error of 0.5% means the 100 gram sample has an actual mass between 99.5 grams± and 100.5 grams. You will need to ﬁnd the rate ﬁrst and then the half-life.) d. If the mass of a radioactive sample can only be measured to a precision of 0.0005%, what range of values for the half-lifes does this imply for a 100± gram sample which decays to a 98.7 gram sample after 1 year? e. In a more recent and more accurate experiment the half-life of 44Ti is reported to be 59.2 0.6 years for a 5 year experiment. Using a half-life of 59.2 years, how much± of a 100 gram sample would decay in 5 years?

10. Biodiversity, or all the various forms of life that currently exist on the earth, is being intensively studied. Biodiversity may include species diversity, genetic diversity, and ecological diversity. Many scientists and lay people believe that the earth’s biodiversity is in jeopardy due to the 6.6. HOMEWORK PROJECTS: 137

marked increase in the rate of species extinction. The purpose of this project is to familiarize you with one measure of species diversity that uses logarithms in its calculation. Species diversity is a community characteristic that describes structure within a group of organisms. A community has a low species diversity if only a few species are present, or if only a few species are abundant. In contrast, a community exhibits a high species diversity if many species are present and they occur in equal or nearly equal numbers. A high species diversity often indicates a highly complex community with many interspecific interactions. Some scientists also believe that high species diversity results in high community stability (ability to be unaffected by disturbance); however, this notion is subject to considerable debate. The simplest measure of species diversity is species richness. Species richness is a count of the total number of species in an area and doesn’t consider the relative abundance of each species in the community. More elaborate measures of species diversity consider both the total number of species in an area as well as the number of individuals of each species. These indices are based on information theory and they were first applied to community analysis by MacArthur (1955) and Margalef (1958) (See Appendix B). The basic premise behind the “uncertainty” concept of information theory is that if a community has a low species diversity, you can be relatively certain of the identity of the species when one is selected at random from the community. In contrast, it would be much more difficult to predict the identity of an individual randomly selected from a highly diverse community. The Shannon diversity index is one of the most widely-used calculations in the ecological literature. The equation for the Shannon diversity index is:

K H0 = p ln p − i i Xi=1

where K is the number of diﬀerent species, pi = ni/N where ni is the number of individuals within species i, and N is the total number of individuals of all species. The symbol means to add up all of the terms following the summation symbol. P For example

5 i Xi=1 means to sum 1+2+3+4+5. Notice that the number under the summation symbol tells you at what index number you should start with and the number above the summation symbol tells you what number to end with. 138 CHAPTER 6. LOGARITHMS

If the sum is

7 pi Xi=1

then you are to find p1 + p2 + p3 + p4 + p5 + p6 + p7. a. Calculate the Shannon diversity index for the Native Prairie and the Shannon diversity index for the Improved Pasture using the data in Ta- ble 6.7. Compare these two indices. Which habitat is more diverse? b. Obtain a “bug” net, kill jar, and zip-lock bag from your instructor. Select two different study sites in which you would expect to find different communities of “bugs” (e.g., pasture and wooded area; lawn and roadside ditch). To collect “bugs” in the first area, make a sweeping motion of the net back-and-forth in front of you as you walk 30 paces. As you collect bugs in your net, carefully drop them into the kill jar. Remove the bugs and place them in a zip-lock bag. After you are done collecting your bugs, spread the dead bugs out on a table and separate them into groups of similar-looking bugs. Identify the number of different types of bugs you have (e.g., moths, beetles, ants) and the number of individuals of each type and use this information to make a chart similar to the chart in part a. Repeat this procedure for the second habitat. Use this data to calculate the Shannon diversity index for your two bug communities. What does the Shannon diversity index tell you? How many types did you have in each habitat? Were the communities of bugs in the two areas similar? Explain why the two communities of bugs were similar or different. Do you think your sample was a good indicator of the bugs present in each community?

Species Native Prairie Improved Pasture Hispid Cotton Rat 9 44 Deer Mouse 7 11 House Mouse 2 4 Fulvous Harvest Mouse 16 0 Least Shrew 4 0 Elliot’s Short-tailed Shrew 3 0 Hispid Pocket Mouse 0 5 Total Number of Captures (N) 41 64

Table 6.7: Number of small mammals captured in native prairie and improved pasture by species Chapter 7

Systems of Equations

7.1 Objectives:

After completing this chapter, you should be able to

Solve two linear equations with two variables • Solve three linear equations with three variables • Solve two nonlinear equations with two variables •

7.2 Motivation:

In many “real-world” situations, we have more than one unknown variable and more than one equation describing the relationship between the variables. The collection of these equations is called a system of equations or simultaneous equations. The Egyptians, sometime around 1950 B.C., were able to solve two equations with two unknowns while the Babylonians in about 1600 B.C. were able to solve systems with 10 equations and 10 unknowns. A collection of 246 problems that was written by the Chinese sometime between 206 B.C. and 222 A.D. contained systems of linear equations. Likewise the Greeks in about 500 A.D. compiled 46 number problems that contained systems of 4 equations and 4 unknowns. Today, systems of equations occur in the description of a wide array of practical situations. In chemistry, systems of equations can be used to describe the transport of chemicals through the body and to design a model to regulate sulfur emissions. In physics, systems of equations are used to help design supercomput- ers and to design reentry simulations for the spacecraft. In biology, systems of equations are used to model the environment and in developing new blood tests. The equations that are used in these applications are quite complex and involve advanced mathematics to solve. In this chapter, we will restrict our attention

139 140 CHAPTER 7. SYSTEMS OF EQUATIONS to just two or three equations, which will be linear, quadratic, exponential, or logarithmic, and two or three unknowns.

7.3 Demonstration:

In the Introduction to Chapter 4, we discussed the different equations that describe velocity and acceleration. In all, we had three equations: v + v v = 0 2 where v was the average velocity, v was the final velocity, and v0 was the initial velocity; y y v = − 0 t where y was the final position, y0 was the initial position, and t was the elapsed time; and v v a = − 0 t where a was the average acceleration. We are going to use these three equations to derive the formula 1 y = at2 + v t + y 2 0 0 which is the equation that we used to describe the location of an object at time t. We will use the Method of Substitution to derive the third equation.

Exploration Question #1: If we examine the formula that we wish to derive, we notice that this formula does not contain the variable v or v. Thus, in deriving our final formula, we want to eliminate these two variables. Because of the structure of the first two equations, it is easiest to first eliminate v. Use the first two equations to derive a formula for v that does not contain v.

Exploration Question #2: Now we wish to use the formula you derived in Exploration Question #1 and the third formula to eliminate the variable v. Substitute the formula that you derived in Ex- ploration Question #1 for v into the third formula and solve for y. 7.4. THE MATHEMATICS: 141 7.4 The Mathematics: 7.4.1 Two Linear Equations A system of equations is merely two or more equations that we are interested in solving simultaneously. A solution to a system are values for the variables that satisfy every equation in the system. If we have two equations and two unknowns in the system of equations, we can graph the equations in a Cartesian coordinate system. In this case, a point of intersection of the two graphs will be the solution of the system as it is a point that simultaneously satisﬁes both equations. When the equations are lines in a system of equations, we say that we have a linear system of equations. In this section we will be interested in two equations and two variables. We can then graph the equations in a Cartesian coordinate system. A solution of the system will be a point of intersection of the two equations. If a system of equations has a point of intersection it is called consistent. When we graph two lines, we see that we have three cases for the possible intersection of the two lines. The ﬁrst possibility is that the lines are parallel - in this case the lines do not intersect and we would have no solution to the system. This case is referred to as inconsistent. 142 CHAPTER 7. SYSTEMS OF EQUATIONS

The second possibility is that we have the same line twice - if we graphed both of the lines, they would be the same line. There are an inﬁnite number of points that lie on both lines so we would have an inﬁnite number of solutions to the system. This case is called dependent.

The third possibility is that the lines do intersect in one point. In this case, we would want to ﬁnd the x and y coordinate of the point of intersection.

There are two methods used to solve systems of linear equations: the method of substitution and the method of elimination. We will do two examples for each method. Method of substitution: In this method, we take one of the equations and solve for one of the variables. We will then use this to substitute into the second equation to reduce it to one variable. Example 1: Suppose that the two equations are

x 2y =3 − and y 3x = 14. − − Although it doesn’t matter which equation you start with and which variable you solve for, sometimes one equation will be easier to work with than the other. For this example, we can solve the ﬁrst equation for x to get

x =3+2y. 7.4. THE MATHEMATICS: 143

We will substitute 3 + 2y in for x into the second equation to get

y 3(3+2y) = 14. − − We now have one equation with just one variable and we can solve this equation for the remaining variable. Distributing the 3 gives us

y 9 6y = 14 − − − and combining like terms gives us

5y = 5. − − Dividing both sides by -5 we see that

y =1.

Once we have found one of the variables, we will return to the first equation to find the second variable. When we solved the first equation for x we got

x =3+2y.

Now that we know y =1, we can substitute this in for y to ﬁnd that

x =3+2(1)=5.

Thus the point of intersection for this system is (5,1). Example 2: Now suppose that the two equations that we are solving are

y =7 3x − and 9x +3y = 21. Since the ﬁrst equation is already solved for y, we can go ahead and substitute 7 3x into the second equation for y. This give us − 9x + 3(7 3x)=21. − Simplifying, we get 9x +21 9x = 21 − or 0=0. In this case, everything canceled out leaving us with a true statement that no longer involves any variables. When this happens, the two equations are really just describing the same line and so there are an inﬁnite number of solutions. This is a dependent system. 144 CHAPTER 7. SYSTEMS OF EQUATIONS

Method of elimination: To solve a linear system using elimination, we want to arrange the equations so that when we add two of the equations, one of the variables cancels out or is eliminated. Example 3: Suppose that the two equations are

2x y =3 − and 3x +4y = 1. − To solve this system by elimination, we want to multiply one of the equations by a constant so that when we add the two equations, one of the variables cancels out. In this example, it is probably easiest to multiply the ﬁrst equation by 4 so that the y terms of the two equations will cancel out. Multiplying each term of the ﬁrst equation by 4, gives us the new equation

8x 4y = 12. − Adding this equation with our second equation gives us

8x 4y = 12 − 3x +4y = 1 − 11x = 11 Solving this for x, we get x =1. We can now substitute x = 1 into either of the original equations to get that y = 1. Thus the point of intersection of this equation is x =1, y = 1 or written as− an ordered pair, (1, 1). − − Example 4: Suppose that our two equations are

2y 4x =7 − and 3y 15=6x. − In this example, we first want to get the x and y terms on one side and the constant on the other. Moving the 6x onto the left hand side of the second equation and the 15 onto the right hand side, we have − 2y 4x =7 − and 3y 6x = 15. − Comparing the coefficients on the x terms of both equations and the y terms of both equations, we notice that multiplying just one of the equations by a constant will not be enough to eliminate one of the variables. If we want to 7.4. THE MATHEMATICS: 145 eliminate the y terms, we have to find a least common multiple of the coefficients of the y terms, in this case 6. So we will multiply the first equation by 3 to get

6y 12x = 21 − and the second equation by -2 to get

6y +12x = 30. − − We multiplied the second equation by 2 so that when we add the two equations the terms will cancel. − Adding these two equations, we have

6y 12x = 21 − 6y + 12x = 30 − − 0y +0x = 9 − or that 0 = 9. In this case,− all the terms with variables canceled leaving us with an obviously false statement. If this happens, there are no solutions to the system, and we say the system is inconsistent. Remember, this occurs when the two lines are parallel and have no point of intersection. Exercises : Solve the following systems of equations. If the system does not have a single solution, state whether the system is inconsistent or dependent.

1. 2x + y = 1 and y = 2x +5 − 2. 3u 2v =5 and 4v =6u 10 − − 3. s + t =1 and 2s +3t =3 4. 2x + y =6 and 3x 4y = 12 − 5. 7x 2y =2 and 5x +4y = 42 − − 6. 3x 8y =1 and 5x +2y = 14 − 7. 9x 6y = 3 and 6x +4y = 2 − − − 146 CHAPTER 7. SYSTEMS OF EQUATIONS

7.4.2 Three Linear Equations Solving a system of linear equations that involves three equations and three unknowns is similar to solving a system of linear equations that involves two equations and two unknowns. The ultimate goal is to reduce the system to one equation with one unknown. Although the method of substitution that we used in the previous section will also work for three or more equations, in some cases it is easier to solve a system using the method of elimination instead. We will solve one system using substitution and one using elimination. Example 1: Suppose that you need to ﬁnd the solution to the following system:

x 2y +3z = 2 − − 4x +10y +2z = 2 − − 3x + y +10z =7 The first thing that we might notice is that in the second equation every coefficient is divisible by 2. When this happens, it will always be easier to divide each of the coefficients by the common factor to reduce the size of the numbers that we will be working with. If we divide each coefficient in the second equation by 2 we get

x 2y +3z = 2 − − 2x +5y + z = 1 − − 3x + y +10z =7. We can solve the ﬁrst equation for x to get

x = 2+2y 3z − − and we can substitute this in for the variable x into the second and third equations. For the second equation this would give us

2( 2+2y 3z)+5y + z = 1. − − − − Combining like terms, we get

y +7z = 5. − For the third equation, substituting 2+2y 3z in for x gives us − − 3( 2+2y 3z) + y +10z =7. − − Combining like terms, we get

7y + z = 13. 7.4. THE MATHEMATICS: 147

Notice that we have reduced the system of three equations of three unknowns to the following system of two equations and two unknowns:

y +7z = 5 − 7y + z = 13 We now can continue to solve this new system as we did in the previous section. We can solve the ﬁrst equation for y to get

y = 5 7z − − and we can substitute this into the equation 7y + z = 13 to get

7( 5 7z) + z = 13. − − Solving for z gives us that z = 1. We can use this value for z to get a value for y and for x. Since y = 5 −7z and z = 1, we see that − − − y = 5 7( 1) = 5+7=2. − − − − Since x = 2+2y 3z and z = 1 and y =2, we see that − − − x = 2 + 2(2) 3( 1) = 2+4+3=5. − − − − So the solution to this system of equations is x = 5, y = 2, and z = 1. We could also write this as (5, 2, 1). − − Example 2: To solve the previous system of

x 2y +3z = 2 − − 2x +5y + z = 1 − − 3x + y + 10z =7 by elimination, we would start by trying to ﬁgure out how to eliminate one of the variables. If we multiply the ﬁrst equation by 2 we get

2x 4y +6z = 4. − − Combining this with the second equation, we see that if we add these two equations together the terms containing the variable x will be eliminated. Adding, we have

2x 4y +6z = 4 − − 2x +5y + z = 1 − − y +7z = 5 − 148 CHAPTER 7. SYSTEMS OF EQUATIONS

Now we want to use the first original equation and the third equation to eliminate the x value. Since the coefficient of x in the third equation is a 3, we should multiply− the first equation by -3. Doing this we get 3x +6y 9z =6. − − Adding this equation with the third equation, we have 3x +6y 9z =6 − − 3x + y +10z =7 7y + z = 13 We now have two equations with two variables. If we were to continue to solve this by elimination we could multiply the equation y +7z = 5 by -7 since the coefficient of y in the equation is 7y + z = 13 is 7. We would− now have 7y 49z = 35. Adding this equation to 7y + z = 13 we have − − 7y 49z = 35 − − 7y + z = 13 48z = 48 − or that z = 1. We can now back substitute to find the rest of the values for x and y. − Substituting z = 1 into 7y + z = 13, we have − 7y 1=13. − Solving for y, we have 7y = 14 so that y = 2. Now we can substitute z = 1 and y = 2 into one of the original equations to find x. Using the first equation− we have x 2(2) + 3( 1) = 2 − − − or that x =5. Thus the solution (as we saw earlier) is (5,2,-1). Exercises : Solve the following systems of equations. If the system does not have a single solution, state whether the system is inconsistent or dependent. 8. 2x + y z = 3; x y z = 0; x + y +2z =5 − − − − 9. r +2s +3t = 3; 2r + s t = 4; r s +4t =7 − − − − 10. 2u v + w = 6; 3u +2v w = 8; u 4v 5w = 44 − − − − − − 11. 2x 3y +2z =5; 4x +6y z =3; 8x +3y 3z = 3 − − − − 12. 5x + y 2z = 10; 2x 3y + z = 6; x +2y z =4 − − − 13. 2u 3v + w =0; 3u + v w = 6; 4u 2v +2w = 8 − − − − − 7.4. THE MATHEMATICS: 149

7.4.3 Nonlinear Systems If the equations in a system of equations are not all linear, we say that we have a nonlinear system of equations. For two equations, we could have a wide range of possibilities. For example, one equation could be the equation of a parabola and the other could be the equation of a line, or, both equations could be exponential. For some nonlinear systems, it is impossible to find an algebraic solution (for example, a line and an exponential). In this case, you can graph each equation and approximate the solution to the equation numerically. In this section, we will discuss nonlinear systems which can be solved algebraically. Since it would be impossible to discuss every single possibility, we will just work through some of the more common cases. In each case, we first try to reduce the two equations with two variables down to one equation with one variable. We then can solve the one equation with the one variable. The ideas that we use to solve these systems should be applicable to other systems as well. For our first case, suppose that one equation is the equation of a line and the other equation is the equation of a parabola. A solution to this system will also be a point of intersection of the graphs of the line and the parabola. If we examine the situation graphically, we see that there are three possibilities as shown in the graphs below. No Solutions:

One Solution: 150 CHAPTER 7. SYSTEMS OF EQUATIONS

Two Solutions:

So, we could either have no solution, one solution, or two solutions to a system containing one line and one quadratic. Example 1: As an example of how to solve such a system, consider the system of equations given by y = x2 +5x +5 y =2x +3. Remember, our goal is to get one equation with one variable. In this system, we can eliminate the variable y by setting the two equations equal to each other - since each is equal to y, they must be equal to each other. Doing so, we get

x2 +5x +5=2x +3

Combining like terms and getting a 0 on one side we have

x2 +3x +2=0.

This is a quadratic in x and we need to either factor or use the quadratic formula. For this equation we can factor to get

(x + 2)(x +1)=0 so either x +2=0or x +1=0. Thus, we see that x = 2 or x = 1. We now need to ﬁnd the corresponding y values.− We can ﬁnd− these by substituting the x values into either original equation. However, it is probably easier to substitute the values into the equation y =2x+3. Substituting x = 2, we get − y = 2( 2)+3= 4+3= 1, − − − so one solution to the system is (-2,-1). Substituting x = 1 into y =2x + 3 we get − y = 2( 1)+3= 2+3=1 − − so the other solution to the system is (-1,1). 7.4. THE MATHEMATICS: 151

Example 2: For another example, suppose that the system of equations consists of two exponential equations:

y = e0.01t

y = 10e0.2t. Again, we ﬁrst want to get one equation with one unknown. Since each equation is equal to the variable y, we can set the two equations equal to each other to get e0.01t = 10e0.2t. We now have to remember how to solve equations involving exponentials. First, we need to get one exponential on one side. To do this, we can divide both sides by e0.2t to get e0.01t = 10. e0.2t From our properties of exponents, we have that

0.01t e 0.01t 0.2t 0.19t = e − = e− . e0.2t This reduces the above equation to

0.19t e− = 10.

We can now take the natural logarithm of both sides to get

0.19t ln e− = ln10.

Using the properties of logarithms, we have

0.19t lne = ln10. − Since ln e =1, we see that ln10 t = 12.12. 0.19 ≈ − − Once we have found t, we can substitute into either original equation to ﬁnd y. Using the equation y = e0.01t, we see that

0.01( 12.12) y e − 0.89. ≈ ≈ So the solution to this system is approximately (-12.12,0.89). 152 CHAPTER 7. SYSTEMS OF EQUATIONS

Exercises : Solve the following systems of equations. If the system does not have a ﬁnite number of solutions, state whether the system is inconsistent or dependent.

14. y = x2 +3x 5 and y = x2 +4x +6 − 15. 2x + y = 1 and y = x2 +3x 5 − 16. y = e3x and 4y = e0.2x 17. x2 + y2 = 4 and x2 4x + y2 =0 − 18. x2 + y2 = 9 and x2 + 1 = y 19. y = x2 4 and y = x +2 − 20. x = e3y and y = ln x +3 7.5. HOMEWORK PROJECTS: 153 7.5 Homework Projects:

1. The United States Standard Atmosphere is displayed below as a function of altitude (km) versus pressure (mb) and as a function of temperature (◦C) versus pressure (mb). Determine the equation of the line for both graphs. Use these two equations to determine the equation that describes the temperature as a function of altitude. Graph this new equation. Using that equation, ﬁnd the temperature at altitudes of 3 km and 7 km. At what altitude would you expect the temperature to be 0 degrees Celsius? 154 CHAPTER 7. SYSTEMS OF EQUATIONS

2. There are three main temperature scales, Fahrenheit, Celsius, and Kelvin all named after the scientists who originated the scales. In certain con- texts, one scale might be more appropriate to use than another scale and there are linear formulas that allow you to convert from one system to another. To convert from Celsius to Fahrenheit, the equation is 9 F = C + 32, 5 where F stands for the temperature measured in Fahrenheit and C stands for the temperature measured in Celsius. Is there a temperature T so that the temperature measured in degrees Fahrenheit is equal to the temperature measured in degrees Celsius? To find this temperature (if it exists) we need to solve the system of linear equations 9 F = C + 32 5 and F = C. Graph these two lines with Celsius as the x coordinate and Fahrenheit as the y coordinate. Solve this system (if possible).− If there is a temperature that− is the same in both Celsius and Fahrenheit label it on your graph. If it is not possible to solve this system, explain why it is impossible to find a point of intersection. To convert from Celsius to Kelvin, the equation is K = 273+ C where K stands for the temperature measured in Kelvin and C stands for the temperature measured in Celsius. Is there a temperature T so that the temperature measured in degrees Kelvin is equal to the temperature measured in degrees Celsius? To find this temperature (if it exists) we need to solve the system of linear equations K = 273+ C and K = C. Graph these two lines with Celsius as the x coordinate and Kelvin as the y coordinate. Solve this system (if possible).− If there is a temperature that− is the same in both Celsius and Kelvin, label it on your graph. If it is not possible to solve this system, explain why it is impossible to find a point of intersection. To convert from Kelvin to Fahrenheit, we need to combine the equation 9 F = C + 32, 5 7.5. HOMEWORK PROJECTS: 155

with the equation K = 273+ C to get a formula involving just Kelvin and Fahrenheit. Solve the second equation K = 273+ C for C and substitute this into the equation for Fahrenheit for C. After simplifying, you should have an equation involving only F and K. Using this new equation, we can ask the question of whether or not there is a temperature T so that the temperature measured in degrees Fahrenheit is equal to the temperature measured in degrees Kelvin? To ﬁnd this temperature (if it exists) we need to solve the system of linear equations using the equation you found involving F and K and the equation F = K. Graph these two lines with Kelvin as the x coordinate and Fahrenheit as the y coordinate. Solve this system (if possible).− If there is a temperature that− is the same in both Kelvin and Fahrenheit label it on your graph. If it is not possible to solve this system, explain why it is impossible to ﬁnd a point of intersection.

3. This project is an example of Pascal’s Mystic Hexagon problem. Graph the parabola y = x2 on a sheet of graph paper. We are going to make a hexagon (a six-sided figure) by picking 6 points on the parabola and connecting the points with line segments. For the six points, use the following x coordinates. For each of the points you need to find the corresponding y− coordinates and label these points on your graph. For point A, let x =− 3; for point B, let x = 1; for point C, let x = 0; for point D, let x =− 2; for point E, let x = 3;− and for point F, let x = 4. Once you have labeled the points A-F on your graph, connect point A and point B with a line segment, connect point B and point C with a line segment, connect point C and point D with a line segment, connect point D and point E with a line segment, connect point E and point F with a line segment, and connect point F and point A with a line segment. You should have a 6 sided figure called a hexagon. We now are going to look at the intersection of “opposite” lines. For the first case, you need to find the intersection of the line through point A and point B with the line through point D and point E. To do this, you need to find the equations of these lines. Once you have found the equations for these two lines, find their point of intersection (or in other words, solve this system of equations). On your graph extend these two line segments and mark the point of intersection on your graph. Label this point of intersection i on your graph. For the next case, we want to find the intersection of the line through point B and point C with the line through point E and point F. To do this, you need to find the equation of the line through point B and point C and the equation through point E and F. Once you have found these equations for these two lines, find their point of intersection. On your 156 CHAPTER 7. SYSTEMS OF EQUATIONS

graph extend these two line segments and mark the point of intersection on your graph. Label this point of intersection ii on your graph. For the last case, we want to find the intersection of the line through point C and point D with the line through point F and point A. To do this, you need to find the equations of these two lines. Once you have found these equations for these two lines, find their point of intersection. On your graph extend these two line segments and mark the point of intersection on your graph. Label this point of intersection iii on your graph. You should now have on your graph the three points of intersections marked that you found from solving the system of equations. If you have graphed your lines and parabola neatly enough, these three points should look like that they are all on the same line. To verify that they are all on the same line, find the slope between the first (i) and second (ii) point, the second (ii) and third point (iii), and the first (i) and third (iii) point. What should be true about the slopes if these points are all on the same line? 4. In electronics there are a pair of axioms that describe circuits with batter- ies and resistors. (A resistor can be any electrical appliance, a light bulb, a TV, a radio, etc.) These axioms are called Kirchoff’s rules. The first rule represents conservation of energy. The amount of energy any charge gains in the process of traversing a circuit must be exactly balanced by the amount of energy the charge loses. The energy per charge is called potential (or voltage). The second rule represents conservation of charge. If a given amount of charge is flowing in a circuit over a time interval (we call this current) and the circuit forks, then some of the current will go into each branch of the fork. The sum of the currents exiting the fork will equal the sum of the currents entering the fork. In electronics the fork is referred to as a node or a junction. Algebraically each complete circuit or loop yields an equation and each node yields an equation.

The above circuit can be split up into three loops and two nodes. The three loops and their corresponding equations are: 7.5. HOMEWORK PROJECTS: 157

Loop 1:

9 3I +4 2I =0 − 3 − 2 Loop 2:

9 3I 6I =0 − 3 − 6 Loop 3:

2I 4 6I =0 2 − − 6 158 CHAPTER 7. SYSTEMS OF EQUATIONS

The two nodes and their equations are: Node A: I3 = I2 + I6

Node B: I2 + I6 = I3

Since there are only three unknowns only three of the ﬁve equations are needed. Notice that the two node equations are really the same, so only use one of the node equations. Pick any two of the loop equations to obtain the three equations necessary. Now solve this system of equations and ﬁnd the currents in each of the three branches.

5. In elastic collisions the kinetic energy and momentum of the objects involved are conserved. In most areas of science the word “conserved” means 1 “constant.” The kinetic energy of an object is 2 the product of its mass and the square of its velocity. The momentum is the product of the mass and velocity. In this project we are going to answer the following question: “If a 10 kg ball is initially at rest and is hit elastically by a 5 kg ball moving at 20 m/sec, how fast are both balls moving afterwards?” a. What is the initial momentum for the ball initially at rest? What is the initial momentum for the ball moving at 20 m/sec? If you add these two momentums together, what is the initial momentum for both balls? b. The final momentum for a ball is given by the formula m v where m · f is the mass of the ball and vf is the final velocity of the ball. Since we have two balls, let the final velocity of the first ball be denoted by v1f and the final velocity of the second ball be denoted by v2f . If we add the two momentums together, we get the equation

P = m1v1f + m2v2f .

Substitute the values for m1 and m2 into this equation. We do not know the values for v1f and v2f at this time, but will use a system of equations to solve for the velocities. c. What is the initial kinetic energy for the first ball? What is the initial kinetic energy for the second ball? If you add these two kinetic energies together, what is the initial kinetic energy for the two balls? d. The final kinetic energy for the first ball is given by the formula KE = 1 2 2 m1v1f and the final kinetic energy for the second ball is given by the 1 2 formula KE = 2 m2v2f . Substitute in the values for m1 and m2 and add these two formulas together to get the final kinetic energy for both balls. e. Since momentum is conserved, the equation from part a and the equation from part b must be equal to each other. Set these two equations 7.5. HOMEWORK PROJECTS: 159

equal to each other for your ﬁrst equation in this system. Likewise, kinetic energy is conserved, so the equation from part c and the equation from part d must be equal to each other. Set these two equations equal to each other for your second equation in this system. You should now have two equations in two unknowns, v1f and v2f .

f. Solve the system of equations for v1f and v2f . Interpret your answers in terms of the balls with which we started this problem. 6. In many universities, faculty members are evaluated on their performance in research, service, and teaching and these evaluations are used to determine pay increases. The department chair is given a set amount of allocation, say $1000 in this example, that she may allocate to the faculty according to their evaluation. To simplify this problem, suppose there are three faculty members eligible for merit pay increases and their evaluations are as follows: It is decided within the department that someone whose evaluation is good deserves twice as much as someone whose evaluation is satisfactory and that someone whose evaluation is outstanding deserves four times as much as someone whose evaluation is satisfactory. Let x be the amount of increase a faculty member should receive for a satisfactory evaluation in research, let y be the amount of increase a faculty member should receive for a satisfactory evaluation in teaching, and let z be the amount of increase a faculty member should receive for a satisfactory evaluation in service. Then, for example, the ﬁrst faculty member should receive a 4x +2y + z increase.

Faculty member Teaching Research Service 1 Good Outstanding Satisfactory 2 Good Good Satisfactory 3 Outstanding Satisfactory Outstanding

a. Find expressions for the amount of increase faculty members 2 and 3 should receive. b. Since the total amount of increase the chair has to distribute is 1,000 dollars, the sum of the three raises should equal 1,000. Use this fact to combine the above three equations into one equation involving x, y, and z. c. The department decides that teaching should be given the highest priority with research being the second highest priority and service being the third highest priority. They decide that the teaching should be compensated at three times the rate for service (or that y = 3z) and that research should be compensated at two times the amount for service (or that x =2z). Use these equations and the equation you got in part b, to solve this linear system of equations. 160 CHAPTER 7. SYSTEMS OF EQUATIONS

d. Determine the amount of raise that each of the three faculty members will receive. e. To determine the faculty member’s overall rating a numerical score of 1 is given for each ‘satisfactory’, a 2 is given for each ‘good’, and a 3 is given for each ‘outstanding’. The overall rating would be the sum of the three ratings assigned by this scheme. Is it possible under this system for a faculty member to receive an overall lower rating but still receive a higher pay raise? If so, give an example of two faculty members, one whose overall rating is higher but who gets less of a pay raise than the second. If it is not possible, explain why. 7. A thermistor is an electrical device that changes electrical properties with changes in temperature in a predictable manner. It is used to measure extreme temperatures where a conventional thermometer might freeze or break. It is also used to control electrical devices that are dependent on temperature control (such as an air conditioner.) a. In the following chart, the temperature in degrees Celsius is given in relation to the variable R, which is measured in the 1000’s of Ohms. Using technology ﬁt a line to this data with T as the x variable and R as the y variable. − − Temperature R (1000’s of Ohms) -10 15 0 21 10 29 20 38 30 47 40 59 50 67

b. Ohm’s Law is an empirical relationship relating electric potential (V ) to current (I). The proportionality constant (R) is called the resistance. This formula is V = RI. If the voltage in a circuit containing a thermistor is 20 volts and the current in the circuit is 0.0005 Amps, find the temperature of the thermistor. You will need to use the equation that you found in part a) as well as Ohm’s Law to find the temperature. c. If the voltage is fixed at 5 volts, combine the equation you got in part a with Ohm’s Law to write an equation containing only the variables T and I. Solve this equation for T. Graph this equation on graph paper with I as the independent variable and T as the dependent variable. 7.5. HOMEWORK PROJECTS: 161

8. In this project we are going to compare the rate of growth of food production versus the rate of growth of population. This project originally appeared in Primus and was written by Michael McDonald, Emily Puck- ette, and Charles Vuono (See Appendix B). In his “Essay on the Principle of Population” (1798), Thomas Malthus stated

“...I say, that the power of population is indeﬁnitely greater than the power in the earth to produce subsistence for man. Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio. A slight acquaintance with numbers will shew (sic) the immensity of the ﬁrst power in comparison of the second.”

UN Year Population Food Production 0 14.02 4.26 10 15.40 4.80 20 17.10 5.42 30 18.95 6.00 40 20.65 6.57 50 23.05 7.10 60 25.43 7.71 70 28.00 8.27 80 31.25 8.84 90 34.25 9.44 100 37.86 10.01

Table 7.1: Population and Food Production

b. Find an exponential function that expresses total population as a function of time. c. Graph (on graph paper) the two functions that you found in parts a and b on the same axes. d. From these two functions and your graph, estimate when population growth will overpower food production. Will famine strike the country described in Table 7.1? When? Perhaps it already has! e. Discuss in writing this modeling procedure in light of the real world: Is it reasonable to assume a linear food production model? What would exponential population growth really mean, and is it reasonable to assume exponential population growth? Are there any other weaknesses and lim- itations in this modeling process?

9. In this project we are going to compare the rate of growth of food production versus the rate of growth of population. This project originally appeared in Primus and was written by Michael McDonald, Emily Puck- ette, and Charles Vuono (See Appendix B). In his “Essay on the Principle of Population” (1798), Thomas Malthus stated

Simply put, this statement claims that food production is a linear function with respect to time and that population growth is an exponential function with respect to time. The year is UN year 100. Data on population (reported in millions) and food production (reported in millions of tons) have been gathered since UN year 0 and are found in Table 7.2. a. The UN has deﬁned starvation as any state in which the average individual receives less than 500 pounds of food per year. If each individual receives 500 pounds or more food per year then the population has enough food to be adequately fed. Using the Food Production column from Ta- ble 7.2, make a new column that contains the number of the most people (in millions) that can be adequately fed with the available food. Using this new column, ﬁnd a linear function that expresses the size of the population that can be adequately fed as a function of time. b. Find an exponential function that expresses total population as a function of time. 7.5. HOMEWORK PROJECTS: 163

UN Year Population Food Production 0 2.40 1.70 10 2.95 1.85 20 3.40 2.03 30 4.40 2.12 40 5.30 2.30 50 6.50 2.45 60 7.80 2.63 70 9.75 2.70 80 11.60 2.92 90 14.50 3.05 100 17.65 3.22

Table 7.2: Population and Food Production

c. Graph (on graph paper) the two functions that you found in parts a and b on the same axes. d. From these two functions and your graph, estimate when population growth will overpower food production. Will famine strike the country described in Table 7.2? When? Perhaps it already has! e. Discuss in writing this modeling procedure in light of the real world: Is it reasonable to assume a linear food production model? What would exponential population growth really mean, and is it reasonable to assume exponential population growth? Are there any other weaknesses and lim- itations in this modeling process?

10. In this project we are going to compare the rate of growth of food production versus the rate of growth of population. This project originally appeared in Primus and was written by Michael McDonald, Emily Puck- ette, and Charles Vuono (See Appendix B). In his “Essay on the Principle of Population” (1798), Thomas Malthus stated

Simply put, this statement claims that food production is a linear function with respect to time and that population growth is an exponential function with respect to time. 164 CHAPTER 7. SYSTEMS OF EQUATIONS

The year is UN year 100. Data on population (reported in millions) and food production (reported in millions of tons) have been gathered since UN year 0 and are found in Table 7.3.

UN Year Population Food Production 0 1.46 2.50 10 2.04 3.02 20 2.71 3.41 30 3.75 4.02 40 4.79 4.56 50 6.55 4.99 60 8.65 5.52 70 11.96 5.89 80 16.00 6.37 90 21.24 6.98 100 28.93 7.43

Table 7.3: Population and Food Production

a. The UN has deﬁned starvation as any state in which the average individual receives less than 500 pounds of food per year. If each individual receives 500 pounds or more food per year then the population has enough food to be adequately fed. Using the Food Production column from Ta- ble 7.3, make a new column that contains the number of the most people (in millions) that can be adequately fed with the available food. Using this new column, ﬁnd a linear function that expresses the size of the population that can be adequately fed as a function of time. b. Find an exponential function that expresses total population as a function of time. c. Graph (on graph paper) the two functions that you found in parts a and b on the same axes. d. From these two functions and your graph, estimate when population growth will overpower food production. Will famine strike the country described in Table 7.3? When? Perhaps it already has! e. Discuss in writing this modeling procedure in light of the real world: Is it reasonable to assume a linear food production model? What would exponential population growth really mean, and is it reasonable to assume exponential population growth? Are there any other weaknesses and lim- itations in this modeling process?

11. In this project we are going to compare the rate of growth of food production versus the rate of growth of population. This project originally appeared in Primus and was written by Michael McDonald, Emily Puck- ette, and Charles Vuono (See Appendix B). 7.5. HOMEWORK PROJECTS: 165

In his “Essay on the Principle of Population” (1798), Thomas Malthus stated

UN Year Population Food Production 0 0.89 0.61 10 1.20 1.04 20 1.51 1.37 30 1.85 1.75 40 2.22 2.24 50 2.65 2.60 60 3.20 3.12 70 4.08 3.39 80 4.76 3.93 90 5.90 4.18 100 6.99 4.58

Table 7.4: Population and Food Production

d. From these two functions and your graph, estimate when population growth will overpower food production. Will famine strike the country described in Table 7.4? When? Perhaps it already has! e. Discuss in writing this modeling procedure in light of the real world: Is it reasonable to assume a linear food production model? What would exponential population growth really mean, and is it reasonable to assume exponential population growth? Are there any other weaknesses and lim- itations in this modeling process? 12. The Bureau of the Census has the task of counting all of the people in the United States once every 10 years. It is a massive and costly enterprise involving an ever increasing staff of enumerators. Table 7.5 contains the population of the United States rounded to the nearest hundred thousands and the number of enumerator staff for the years 1790 to 1990. The data is from a Science News article (see Appendix B). Let x = 0 for the year 1790 and using technology, make two scatterplots. The first scatterplot should have the number of years since 1790 as the independent variable and the population of the United States as the dependent variable. The second scatterplot should have the number of years since 1790 as the independent variable and the number of enumerator staff as the dependent variable. Fit an exponential to each of these scatterplots. Discuss how well these equations fit the data. A question has been raised about when the number of enumerator staff will equal the total population of the United States. Use the two equations that you fit to the scatterplots to determine the year in which this will happen. What will be the population of the United States in that year?

Year Pop. of U.S. # of staﬀ Year Pop. of U.S. # of staﬀ 1790 3,900,000 650 1910 92,200,000 70,266 1800 5,300,000 900 1920 106,000,000 87,756 1810 7,200,000 1,100 1930 123,200,000 87,756 1820 9,600,000 1,188 1940 132,200,000 123,069 1830 12,900,000 1,519 1950 151,300,000 142,962 1840 17,100,000 2,167 1960 179,300,000 159,321 1850 23,200,000 3,231 1970 203,200,000 166,408 1860 31,400,000 4,417 1980 226,500,000 458,523 1870 38,600,000 6,530 1990 248,700,000 510,200 1880 50,200,000 31,382 2000 281,422,000 550,000 1890 63,000,000 46,804 2010 308,746,000 635,000 1900 76,200,000 52,871

Table 7.5: Census Data 7.5. HOMEWORK PROJECTS: 167

13. In this project we are going to compare the populations of Mexico and the United States. Table 7.6 contains the population of Mexico rounded to the nearest 10,000 for the years 1980 through 1986 and Table 7.7 contains the population of the United States rounded to the nearest 10,000 for the same years. a. Find the rate of increase of the population of Mexico from 1980 to rt 1986 and use this to write an exponential equation of the form P = P0e (where you have supplied the values for r and P0.)

Year Population 1980 67,380,000 1981 69,130,000 1982 70,930,000 1983 72,770,000 1984 74,660,000 1985 76,660,000 1986 78,590,000

Table 7.6: Population in Mexico

b. Find the rate of increase of the population of the United States from 1980 to 1986 and use this to write an exponential equation of the form rt P = P0e (where you have supplied the values for r and P0.)

Year Population 1980 227,220,000 1981 229,470,000 1982 231,660,000 1983 233,790,000 1984 235,820,000 1985 237,920,000 1986 240,130,000

Table 7.7: Population in United States

c. Using the equations from parts a and b, when will the population of Mexico be the same as the population of the United States? d. Now assume that the population growth for the United States is better described by a linear function. Find a line that best ﬁts the data for the population of the United States population. e. Graph the linear equation for the United States population and the exponential equation for the Mexico population on the same set of axes with time as the independent variable. Use this graph to estimate when 168 CHAPTER 7. SYSTEMS OF EQUATIONS

the population of Mexico will be the same as the population of the United States. f. Discuss in writing the reasonableness of your answers to part c and to part e.

14. The Cornell Nest Box Network began a study in 1997 to study how clutch size (the number of eggs females lay in one nest attempt) varied by geographic location and by year for Eastern Bluebirds. They were interested in studying if clutch size increased from south to north and how the clutch sizes varied from year to year. The data points in Table 7.8 and in Ta- ble 7.9 were estimated from the article Clutch-size Variation in Eastern Bluebirds (see Appendix B). Table 7.8 gives the average clutch size of the bluebirds who had laid their eggs by the indicated lay date (the lay date expresses date as days since January 1) at latitudes of 30◦ N and 45◦ N. Using technology produce two scatterplots for the data in Table 7.8 - one for 30◦ N and one for 45◦ N - with the lay date as the independent variable and the average clutch size as the dependent variable. Fit a quadratic to each scatterplot and discuss how well the equations ﬁt the data set. Describe what is happening to the average clutch size as the lay date increases for each latitude. For the 30◦ N data, when does the maximum clutch size occur for the quadratic model and what is the maximum clutch size? Is there any date in which the clutch sizes are the same? (Hint: You will need to solve a system of equations to answer this question).

Lay Date Average Clutch Size Latitude 60 4.4 30 78 4.5 30 96 4.6 30 114 4.6 30 132 4.5 30 150 4.4 30 166 4 30 180 3.8 30 110 4.8 45 120 4.7 45 135 4.6 45 150 4.5 45 165 4.4 45 180 4.4 45 192 4.3 45 203 4.2 45

Table 7.8: Average Clutch Size by Lay Date 7.5. HOMEWORK PROJECTS: 169

Table 7.9 gives the average clutch size of the bluebirds who had laid their eggs by the indicated lay date for 1997 and for 1998. Using technology produce two scatterplots for the data in Table 7.9 - one for 1997 and one for 1998 - with the lay date as the independent variable and the average clutch size as the dependent variable. Fit a quadratic to each scatterplot and discuss how well the equations ﬁt the data set. Describe what is happening to the average clutch size as the lay date increases for each year. For the 1997 data, what is the maximum clutch size and when does it occur? For the 1998 data, what is the maximum clutch size and when does it occur? Is there any date in which the clutch sizes are the same? (Hint: You will need to solve a system of equations to answer this question).

Lay Date Average Clutch Size Year 60 4.7 1997 77 4.8 1997 94 4.85 1997 111 4.8 1997 128 4.75 1997 145 4.65 1997 162 4.45 1997 175 4.2 1997 60 4.9 1998 77 5 1998 94 5 1998 111 4.85 1998 128 4.75 1998 145 4.6 1998 162 4.3 1998 175 3.95 1998

Table 7.9: Average Clutch Size by Lay Date

15. Table 7.10 below shows the relationship between time spent on a treadmill (in minutes) and total power exerted (in watts). Using technology produce a scatterplot for the data in Table 7.10 with time spent on the treadmill as the independent variable and power as the dependent variable. Does the data look linear or quadratic? Fit a line and a quadratic to this data set and discuss which equation best ﬁts the data set. Table 7.11 shows the relationship between power exerted and oxygen consumption (in liters/min.). Using technology produce a scatterplot for the data in Table 7.11 with power as the independent variable and oxygen as the dependent variable. Fit a line and a quadratic to this data set and discuss which equation best ﬁts the data set. 170 CHAPTER 7. SYSTEMS OF EQUATIONS

Time Power Time Power 1 14 9 128 2 29 10 143 3 43 11 157 4 57 12 171 5 71 13 185 6 86 14 200 7 100 15 214 8 114

Table 7.10: Time (minutes) on treadmill and power exerted (watts)

Power Oxygen Power Oxygen 10 0.48 140 4.55 20 0.94 150 4.69 30 1.37 160 4.82 40 1.78 170 4.91 50 2.16 180 4.99 60 2.53 190 5.03 70 2.86 200 5.06 80 3.18 210 5.06 90 3.18 220 5.03 100 3.18 230 4.99 110 3.18 240 4.91 120 3.18 250 4.82 130 3.18

Table 7.11: Power (watts) exerted vs. oxygen consumed (L/min)

Using the linear equation from the ﬁrst scatterplot and the quadratic equation from the second scatterplot, ﬁnd a quadratic equation with time as the independent variable and oxygen consumed as the dependent variable. Using this new equation, estimate the oxygen consumption for someone that has exercised for 8 minutes, for 15 minutes, and for 30 minutes. Dis- cuss the reasonableness of your answers. Appendix A

Scientiﬁc Notation

Scientific notation is a way of writing numbers that are very large or very small in a more compact form using exponents. For example, numbers such as 456,000,000,000,000,000 and .000000000000000006532 are very cumbersome and tedious to work with. Yet numbers such as these frequently occur in fields as diverse as astronomy, archaeology, biology, chemistry, medicine, or any field where very large or very small quantities are encountered. Scientific notation uses powers of 10 to make these large numbers more manageable. For instance, the numbers in the paragraph above would become 17 18 4.56 10 and 6.532 10− . × × Changing from standard notation to scientific notation:

1. Move the decimal point to the place in the number that will make the number be between 1 and 10.

2. Determine the power of 10 by counting the number of places you had to move the decimal point. If you moved the decimal point to the left then the power is positive, and if you moved it to the right then the power is negative.

Example: Write 93,000,000 in scientiﬁc notation: Solution: Using the two steps above, we have

1. To get a number between 1 and 10 we must put the decimal point between the 9 and 3.

2. This means that our decimal point had to be moved 7 places to the left, thus giving us a positive exponent of 7. Therefore the number in scientiﬁc notation is 9.3 107. × Example: Write .00000000563 in scientiﬁc notation. Solution: Using the two steps above, we have

171 172 APPENDIX A. SCIENTIFIC NOTATION

1. To get a number between 1 and 10 we must put the decimal point between the 5 and 6. 2. This means that our decimal point had to be moved 9 places to the right, thus giving us a negative exponent of 9. Therefore the number in scientiﬁc 9 notation is 5.62 10− . × Changing from scientiﬁc notation to standard notation:

1. Determine if the decimal point needs to be moved to the left or right. This is done by seeing if the exponent is positive (in which case it is moved to the right) or negative (in which case it is moved to the left). 2. Move the decimal the same number of spaces as the power that 10 is raised to in the direction determined in step 1.

8 Example: Write 8.3 10− in standard notation. Solution: Using the× two steps for changing a number from scientiﬁc notation to standard notation, we have

1. The exponent is negative so we will be moving the decimal point to the left. 2. We will move the exponent 8 places to the left. Since there is only one digit to the left of 3 we must add in enough zeros so that it is possible to move the decimal point 8 places. Therefore the number in standard notation is .000000083.

Example: Write 2.55 106 in standard notation. Solution: Using the two× steps for changing a number from scientiﬁc notation to standard notation, we have

1. The exponent is positive so we will be moving the decimal point to the right. 2. We will move the exponent 6 places to the right. Since there are only two nonzero digits to the right of 2 we must add in enough zeros so that it is possible to move the decimal point 6 places. Therefore the number in standard notation is 2,550,000.

Since scientiﬁc notation always involves a base of 10 and space is sometimes limited, some calculators and some computer programs do not actually show the base of 10 but it is still implied. For example, the TI-85 calculator would write 629000000000 as 6.29 E11. The E indicates that the 11 is in the exponent or in other words the number is 6.29 1011. Other types of calculators simply leave a space between the number and× the exponent in which case, the number above would be written as 6.29 11. Still other types of calculators would write this number as 6.2911. Even though this looks like we are raising 6.29 to the 11th power it still really means 6.29 1011. × Appendix B

References and Further Readings

B.1 Chapter 1

B.1.1 Body Mass Index For a comprehensive overview of the current state of the Body Mass Index debate, see the article “The Fat Fracas: Researchers weigh in on body size” by Kathleen Fackelmann in Science News, Vol. 153, pgs. 283-285. To visit Science News online, go to http://www.sciencenews.org.

B.1.2 Empire State Building For information on the Empire State Building, see http://www.esbnyc.com.

B.1.3 Molarity The project on Chemical Solutions was developed by Dr. Diane Dixon (South- eastern Oklahoma State University (SEOSU)), Dr. Brett Elliott (SEOSU), and Dr. Karla Oty (Cameron University). Funding for development was provided during the Summer of 2010 from Oklahoma IDeA Network of Biomedical Re- search Excellence (http://okinbre.org).

B.2 Chapter 2

B.2.1 Bats For general information on bats, see the Bat Conservation International’s website at http://www.batcon.org.

173 174 APPENDIX B. REFERENCES AND FURTHER READINGS

B.2.2 CMR Method

For more information on the CMR Method, see “Research and management techniques for wildlife and habitats (5th edition).” Bookhout, T.A., Editor, The Wildlife Society, Bethesda, Maryland, 1994. For information on applying the CMR Method to estimating animal populations, see Hayne, D.W. 1949. ”Two methods for estimating animal populations.” Journal of Mammalogy, 30: 399-411.

B.3 Chapter 3

B.3.1 Body weight versus ranges

For more information on the relationship between body weight and home range size see McNab, B.K. 1963. ”Bioenergetics and the determination of home range size.” American Naturalist, 97: 133-140.

B.3.2 Target Heart Rates

The project dealing with Target Heart Rates (THR) was originally written by Joseph Myers, Walter Barge, Todd Crowder, and Kathleen Snook and appeared in Interdisciplinary Lively Application Projects (ILAPs). David C. Arney, Edi- tor, 1997: Published and distributed by The Mathematical Association of Amer- ica.

B.4 Chapter 4

B.4.1 History

For a general history of quadratics, see Howard Eve’s book An Introduction to the History of Mathematics, 5th Edition, Saunders College Publishing, 1983.

For a more speciﬁc history of quadratics, see Morris Kline’s three volume series Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972.

B.4.2 Reﬂectivity of water surfaces

The data found in Table 4.2 is from Schalles, et al., 1997, Estimation of algal and suspended sediment loads using hyperspectral sensors and integrated meso- cosm experiments. Proceedings of the 4th International Conference on Remote Sensing for Marine and Coastal Environments, Orlando, FL. B.5. CHAPTER 5 175

B.4.3 Species Diversity

The data found in Table 4.3 is from Collins, S. and D.J. Gibson, 1990, Eﬀects of ﬁre on community structure in tallgrass and mixed-grass prairie. Pp. 81-98 in Fire in North American Tall grass Prairies (S.L. Collins and L.L. Wallace, editors). University of Oklahoma Press, Norman.

B.4.4 Population Dynamics

The project in this chapter dealing with population dynamics was originally written by Brian Alters in Investigating Evolutionary Biology in the Laboratory. William F. McComas, Editor. 1994: Lancaster Press, Lancaster, Pennsylvania. The copyright is held by the National Association of Biology Teachers.

B.5 Chapter 5

B.5.1 History

For a general history of exponentials, see Howard Eve’s book An Introduction to the History of Mathematics, 5th Edition, Saunders College Publishing, 1983.

For an entertaining account of Euler’s life, see E.T. Bell’s book Men of Mathe- matics, Simon & Schuster, Inc., 1965.

B.5.2 Population

The address for the world wide web page of the United States Census Bureau is http://www.census.gov.

B.5.3 Whooping Cranes

The 1938-1984 data on whooping cranes is from Boyce, M.S. 1987. Time-series analysis and forecasting of the Aransas/Wood Buﬀalo whooping crane population. Pages 1-9 in Proceedings of the 1985 Crane Workshop. Whooping Crane Maintenance Trust, Grand Island, Nebraska.

The 1985-1996 data on whooping cranes is from Tom Stehn, Biologist, U.S. Fish and Wildlife Service, Aransas National Wildlife Refuge, Texas. Personal communication to Bryon Clark, Associate Professor, Department of Biological Sciences, Southeastern Oklahoma State University, on November 6, 1997. 176 APPENDIX B. REFERENCES AND FURTHER READINGS B.6 Chapter 6 B.6.1 History For a general history of logarithms, see Howard Eve’s book An Introduction to the History of Mathematics, 5th Edition, Saunders College Publishing, 1983.

B.6.2 Titanium-44 The original articles that described the new measurements for 44Ti are 1. Three-Laboratory Measurement of the 44Ti Half-Life by I. Ahmad, G. Bonino, G. Cini Castagnoli, S.M. Fischer, W. Kutschera, and M. Paul, Phys. Rev. Lett. 80, 2550 (March 23, 1998). 2. Half-Life of 44Ti as a Probe for Supernova Models, J. Gorres et al., Phys. Rev. Lett. 80, 2554 (March 23, 1998).

B.6.3 Shannon Diversity Index For more information on species diversity, see 1. Clark, B.K., B.S. Clark, T.R. Homerding, and W.E. Munsterman. 1998. Communities of small mammals in six grass-dominated habitats of southeastern Oklahoma. Amer. Midl. Nat., 139: 262-268. 2. MacArthur, R.H. 1955. Fluctuations of animal populations and a measure of community stability. Ecology, 36: 533-536. 3. Margalef, R. 1958. Information theory in ecology. Gen. Systems, 3:36-71.

B.7 Chapter 7 B.7.1 History For a general history of systems of equations, see Howard Eve’s book An Intro- duction to the History of Mathematics, 5th Edition, Saunders College Publish- ing, 1983.

B.7.2 Population Growth versus Food Production The projects in this chapter dealing with population growth and linear food production originally appeared in Primus in the article A Precalculus Project on Exponential Population Growth and Linear Food Production by Michael A. McDonald, Emily Puckette, and Charles Vuono. The article can be found in Volume VI, Number 1 (March 1996).

B.7.3 Census data The data for the number of enumerator staﬀ employed by the Bureau of the Census can be found in the article “Census Sampling Confusion” by Ivars Pe- B.7. CHAPTER 7 177 terson in Science News, Vol. 155, Number 10 (March 6, 1999). To visit Science News online, go to http://www.sciencenews.org.

B.7.4 Oxygen Consumption The data for this project appeared in Interdisciplinary Lively Application Projects (ILAPs) and was written by Joseph Myers, Walter Barge, Todd Crowder, and Kathleen Snook. David C. Arney, Editor, 1997: Published and distributed by The Mathematical Association of America.

B.7.5 Bluebirds The data from Table 7.8 and Table 7.9 were estimated from graphs found in Clutch-size Variation in Eastern Bluebirds by Andre A. Dhondt, Tracey L. Kast, and Paul E. Allen. The article appeared in Birdscope, Volume 14, Number 2. For more information on The Birdhouse Network, go to http://nestwatch.org/. 178 APPENDIX B. REFERENCES AND FURTHER READINGS Appendix C

Answers to Odd Problems

C.1 Chapter 1

Exercises :

1. 1,305.7; 1,306; 1,310; 1,300

3. 689,560.5; 689,560; 689,560; 689,600

5. 164.2; 164; 160; 200

7. 3.1; 3; 0; 0

9. Not enough information is given. We need to know how much area one can of paint will cover.

11. Enough information is given; 55 mph ≈ 13. Biased. Students in an algebra class are likely to diﬀer in average age from typical college students.

15. Not biased. Every student has an equal chance of getting a blue marble.

17. Probably biased. Constituents that answer a mail survey are likely to feel more strongly about the issue than typical constituents.

19. .001

21. .05

23. 1,000

25. 190,080

27. 2.54 ≈ 179 180 APPENDIX C. ANSWERS TO ODD PROBLEMS

29. 31,536,000 (using 365 days); 31,556,736 (using 365.25 days) 31. 5,280 33. .447 ≈ 35. 1,000 37. 1,027 ≈ C.2 Chapter 2

Exercises : 1. 6 in2 3. 12 m2 5. 6 ft2 7. 18 in2 9. 27,878,400 ft2 11. 640 acres 13. .015 mi2 15. No. For example, a triangle or a rectangle could have an area of 1 acre. 17. 30 which appears to be pretty close to the true area. ≈ 19. 10 in 21. 15 m 23. (3π +16) ft or 25.4 ft ≈ 25. 42 in or 3.5 ft ≈ ≈ 27. 24 ft3 29. 20π cm3 or 62.8 cm3 ≈ 31. 360 in3 33. .95 liters ≈ 35. 57.8 in3 ≈ 37. The cost per oz for the cans is 2.8 cents. The cost per oz for the bottle is 1.9 cents. Therefore, the better≈ buy is the 2-liter bottle. ≈ 39. 298 ft2 41. a. 28π cm2 or 88 cm2; b. 24π cm2 or 75.4 cm2 ≈ ≈ 43. Answers will vary. They would not necessarily have the same surface area. C.3. CHAPTER 3 181 C.3 Chapter 3

Exercises :

1 1 1 1. a. −2 b. −2 c. −2 d. The slopes are all equal. 3. a. 2

-4 -2 2 4 6 8 10

-5

-10 182 APPENDIX C. ANSWERS TO ODD PROBLEMS

c. 0

-4 -2 2 4

e. -2

-10 -5 5 10

5. negative

7. positive

9. slope-intercept form: y = 5x 5; general form: 5x + y = 5 − − − 11. slope-intercept form: y = 5; general form: y =5

13. slope-intercept form: y = 1; general form: y =1

3 15. 2 17. 0

1 19. −2 21. undeﬁned

3 23. x intercept: (1, 0); y intercept: (0, − ) − − 2 25. x intercept: none; y intercept: (0, 12) − − C.3. CHAPTER 3 183

27. x intercept: (0, 0); y intercept: (0, 0) − − 29. The graph is:

-4 -2 2 4

-2

31. The graph is:

-15 -10 -5 5 10 15 184 APPENDIX C. ANSWERS TO ODD PROBLEMS

33. The graph is:

-4 -2 2 4 -2

-4

-6

35. The graph is:

-4 -3 -2 -1 -1

-2

-3

37. slope: -1; x intercept: (2, 0); y intercept: (0, 2) − − 39. slope: 3; x intercept: ( 1, 0); y intercept: (0, 3) − − − 41. slope: 0; x intercept: none; y intercept: (0, 2) − − − 42.b.1; d.37

43.a.1; c.-2; e.-74

45. It is not a function because for some values of x, there are two possible answers for y.

47. Scatterplot D

49. Scatterplot A

51. Positive linear correlation. We expect that the higher the Math ACT score, the higher the grade would be in algebra. C.4. CHAPTER 4 185

53. Negative linear correlation. We expect that the more police we have, the lower the crime rate will be.

55. -0.92632

57. The x intercept occurs when y is 0. Since y represents the % of 18 to 25 who smoke,− the x coordinate of the x intercept is predicting the year in which no one between− the ages of 18 to− 25 will smoke.

59. 35.8% - Interpolation ≈ C.4 Chapter 4

Exercises :

1. Vertex: ( 2, 17); minimum − − 3. Vertex: ( 2, 7); maximum − 5. Vertex: ( 1, 12); minimum − − 7. y intercept: (0, 4); x intercept(s): ( 4, 0), (1, 0) − − − − 9. (0, 5); x intercept(s): ( 2 √21 , 0) − − − ± 3 11. (0, 5); x intercept(s): none − − 3 25 13. Vertex: ( −2 , −4 ); y intercept: (0, 4); x intercept(s): ( 4, 0), (1, 0) − − − −

-6 -4 -2 2 4 -5 186 APPENDIX C. ANSWERS TO ODD PROBLEMS

15. Vertex: ( 2, 7); y intercept: (0, 5); x intercept(s):( 2 √21 , 0) − − − − − ± 3 10 7.5 5 2.5

-6 -4 -2 2 4 -2.5 -5 -7.5 -10

17. Vertex: ( 4, 19); y intercept: (0, 3); x intercept(s): ( 4 √19, 0) − − − − ±

15 10 5

-10 -8 -6 -4 -2 2 -5 -10 -15

19. Vertex: (1, 3); y intercept: (0, 0); x intercept(s): (0, 0), (2, 0) − − − 15 12.5 10 7.5 5 2.5

-4 -2 2 4 6 -2.5 -5

21. quadratic

23. line

25. The line is the better ﬁt since the adjusted R2 value for the quadratic is lower than the adjusted R2 value for the line. C.5. CHAPTER 5 187 C.5 Chapter 5

Exercises :

1. 81

3. 10,000

5. 20.09 ≈ 7. 0.16

9. y11

11. x5y5

13. x14

15. x11

1 17. x4 19. 1

21. 1

1 23. z3 25. x2

27. Points plotted will vary but both positive and negative values for x should be used.

12.5

7.5

2.5

-4 -2 2 4

-2.5 188 APPENDIX C. ANSWERS TO ODD PROBLEMS

29. The graph is

12.5

7.5

2.5

-3 -2 -1 1 2 3

-2.5

31. No matter what value is substituted in for x, y is always 1. Thus the graph is a horizontal line and is not the graph on an exponential function.

33. The table is

x ln(x) eln(x) 1/8 2.0794 1 − 8 1/4 1.3863 1 − 4 1/2 0.6931 1 − 2 1 0 1 2 0.6931 2 4 1.3863 4 8 2.0794 8 100 4.6052 100

2t 35. y = 20.1e−

C.6 Chapter 6

Exercises :

1. log3 81=4

3. log10 10, 000= 4

5. loga 1=0

2 7. log27 9 = 3 9. 23 =8

11. 91/2 =3 C.6. CHAPTER 6 189

13. 4a = y

3 1 15. 2− = 8

17. by = x

19. 4

21. 4

23. 2 −

25. logb 21

27. 5logb x

ad 29. logb ( c )

31. 1

33. 1 − 35. 1

37. 0.70

39. 1.30

41. 0.69 − 43. The table is

x log2 x 1/8 3 − 1/4 2 − 1/2 1 − 1 0 2 1 4 2 8 3 16 4

The graph is 2

1 2 3 4

-2

-4

-6 190 APPENDIX C. ANSWERS TO ODD PROBLEMS -8

45. They both have x intercepts of (1, 0). One is increasing, the other is decreasing. −

47. t 345.4 ≈ 49. r 138.6 ≈ − 51. t 8.66 ≈ 53. x 148.4 ≈ 55. x 0.027 ≈ 57. x = .3

59. P 1, 343, 705, 000 which is not close at all to the actual population in 1980.≈

61. r 0.013906474. The predicted population in 1980 would be P 237, 685, 000. ≈ ≈ C.7 Chapter 7

Exercises :

1. Inconsistent

3. s =0, t =1

5. x = 2, y = 8 − − 7. Dependent

9. r =0, s = 3, t =1 − 1 2 11. x = 2 , y = 3 , z =3 13. u = 2, v = 2, w = 2 − − − 15. x =1, y = 1; and x = 6, y = 13 − − 17. x =1, y = √3; and x =1, y = √3 − 19. x = 2, y = 0 and x =3, and y =5 − Index

acceleration, 70 factoring, 75 acre, 19 Fahrenheit, 171 area, 19 functions, 52 biodiversity, 152 geometry, 15 bluebirds - eastern, 91, 92, 190 gram, 7 body mass index, 11 gravity, 72, 87, 89 Brahe, Tycho, 90 byte, 35 heart rate - maximum, 66 heart rate - resting, 67 capture-mark-recapture, 17, 41 heart rate - target, 66 CD-ROM, 34 heart rate - target zone, 66 Celsius, 171 hectare, 19 census, 40 home range, 16 circle - area, 20 circle - circumference, 25 inconsistent, 158 circumference, 25 independent variable, 52 clutch size, 91, 92, 190 intensity, 127 consistent, 158 intercepts - line, 48 correlation, 54 intercepts - parabola, 74 cube - surface area, 29 interpolation, 57 cube - volume, 27 Joule, 62 decibel, 127 joule, 33 dependent, 159 dependent variable, 52 Kelvin, 171 Diophantus, 69 Kepler’s Laws, 90 Kepler, Johannes, 90 energy, 31 kinematics, 70 estimation, 1 exponential equations, 135 least-squares method, 54 exponential functions, 99 line, 39 exponential functions - inverse, 109 line - general form, 46 exponential graphs, 107 line - horizontal, 47 exponential growth and decay, 138 line - point-slope form, 46 exponents, 99, 102 line - slope-intercept form, 46 exponents - rules, 103–106 line - vertical, 47 extrapolation, 57 linear regression, 53

191 192 INDEX linear regression - dependent variable, resting energy expenditure, 12 54 right circular cylinder - surface area, 30 linear regression - independent variable, right circular cylinder - volume, 28 55 rounding, 3 logarithm - common, 132 logarithm - natural, 132 sampling, 6 logarithm - rules, 130, 131 slope, 39, 44 logarithmic equations, 137 sound, 126 logarithmic function, 133 species diversity, 87 logarithms, 129 speed, 70 sphere - surface area, 29 meter, 7 sphere - volume, 27 method of elimination, 161, 163 square - area, 19 method of substitution, 159 square - perimeter, 24 metric system, 7, 70 supplement, ii metric system - preﬁxes, 8 surface area, 29 system of equations, 155, 158 Napier, John, 125 system of equations - consistent, 158 newton, 32 system of equations - dependent, 159 Newton’s law of cooling, 121, 149 system of equations - inconsistent, 158 Newton, Isaac, 90 system of equations - linear, 158 newton-meter, 33 system of equations - method of elimination, 161, 163 oxygen consumption, 36, 189 system of equations - method of substitution, 159 parabola, 73, 76 system of equations - nonlinear, 167 perimeter, 24 population growth, 100 technology, ii population size, 40 threshold of hearing, 127 position, 70 threshold of pain, 127 power, 189 triangle - area, 19 preface, i triangle - hypotenuse, 18 pythagorean formula, 20 triangle - perimeter, 24 triangle - right - area, 20 quadratic formula, 75 quadratics, 69 U.S. Census Bureau, 100 unit conversions, 7 r-squared, 54 r-squared - adjusted, 83 velocity, 70 random sample, 6 vertex, 73 range length, 17 volume, 26 rectangle - area, 19 volume - unit conversions, 26 rectangle - perimeter, 24 rectangular solid - surface area, 29 rectangular solid - volume, 27 regression - linear, 53 regression - quadratic, 79