APPENDIX B Mathematical Modeling B1 Appendi B Mathematical Modeling B.1 Modeling Data with Linear Functions Introduction Direct Variation Rates of Change Scatter Plots Introduction The primar objective of applied mathematics is to find equations or mathematical models that describe real-world phenomena. In developing a mathematical model to represent actual data, ou should strive for two (often conflicting) goals accurac and simplicit. That is, ou want the model to be simple enough to be workable, et accurate enough to produce meaningful results. EXAMPLE 1 A Mathematical Model The total annual amounts of advertising epenses (in billions of dollars) in the United States from 1990 through 1999 are shown in the table. (Source: McCann Erickson) Year 1990 1991 199 1993 199 1995 199 1997 1998 1999 19. 17.5 13.7 139.5 151.7 1.9 175. 187.5 01. 15. Advertising e penses (in billions of dollars) 0 00 180 10 10 10 100 80 0 0 0 = 10.19t +11.5 1 3 5 7 8 9 Year (0 1990) t A linear model that approimates this data is 10.19t 11.5, 0 t 9 where represents the advertising epenses (in billions of dollars) and t represents the ear, with t 0 corresponding to 1990. Plot the actual data and the model on the same graph. How closel does the model represent the data? The actual data is plotted in Figure B.1, along with the graph of the linear model. From the figure, it appears that the model is a good fit for the actual data. You can see how well the model fits b comparing the actual values of with the values of given b the model (these are labeled * in the table below). t 0 1 3 5 7 8 9 19. 17.5 13.7 139.5 151.7 1.9 175. 187.5 01. 15. FIGURE B.1 * 11.5 1.7 13.9 17.1 157.3 17.5 177. 187.8 198.0 08.
B APPENDIX B Mathematical Modeling Direct Variation There are two basic tpes of linear models. The more general model has a -intercept that is nonzero: m b, where b 0. The simpler one, m, has a -intercept that is zero. In the simpler model, is said to var directl as, or to be proportional to. Direct Variation The following statements are equivalent. 1. varies directl as.. is directl proportional to. 3. m for some nonzero constant m. m is the constant of variation or the constant of proportionalit. EXAMPLE State Income Ta In Pennslvania, the state income ta is directl proportional to gross income. Suppose ou were working in Pennslvania and our state income ta deduction was $ for a gross monthl income of $1500.00. Find a mathematical model that gives the Pennslvania state income ta in terms of the gross income. State income ta (in dollars) 100 80 0 0 (1500, ) 0 1000 000 3000 000 Gross income (in dollars) FIGURE B. Let represent the state income ta in dollars and represent the gross income in dollars. Then ou know that and are related b the equation m. You are given when 1500. B substituting these values into the equation m, ou can find the value of m. m Direct variation model m 1500 Substitute and 1500. 0.08 m Divide each side b 1500. So, the equation (or model) for state income ta in Pennslvania is 0.08. In other words, Pennslvania has a state income ta rate of.8% of the gross income. The graph of this equation is shown in Figure B..
APPENDIX B Mathematical Modeling B3 Most measurements in the English sstem and the metric sstem are directl proportional. The net eample shows how to use a direct proportion to convert between miles per hour and kilometers per hour. Miles per hour 80 70 0 50 0 30 0 10 FIGURE B.3 (103, ) 0 0 0 80 100 10 Kilometers per hour EXAMPLE 3 The English and Metric Sstems You are traveling at a rate of miles per hour. You switch our speedometer reading to metric units and notice that the speed is 103 kilometers per hour. Use this information to find a mathematical model that relates miles per hour to kilometers per hour. Let represent the speed in miles per hour and represent the speed in kilometers per hour. Then ou know that and are related b the equation m. You are given when 103. B substituting these values into the equation m, ou can find the value of m. m m 103 103 m 0.13 m Direct variation model Substitute and 103. Divide each side b 103. Use a calculator. So, the conversion factor from kilometers per hour to miles per hour is approimatel 0.13, and the model is 0.13. The graph of this equation is shown in Figure B.3. Once ou have found a model that converts speeds from kilometers per hour to miles per hour, ou can use the model to convert other speeds from the metric sstem to the English sstem, as shown in the table. Kilometers per hour 0.0 0.0 0.0 80.0 100.0 10.0 Miles per hour 1..9 37.3 9.7.1 7. NOTE The conversion equation 0.13 can be approimated b the simpler equation 5 8. For instance, to convert 0 kilometers per hour, divide b 8 and multipl b 5 to obtain 5 miles per hour.
B APPENDIX B Mathematical Modeling Rates of Change A second common tpe of linear model is one that involves a known rate of change. In the linear equation m b ou know that m represents the slope of the line. In real-life problems, the slope can often be interpreted as the rate of change of with respect to. Rates of change should alwas be listed in appropriate units of measure. EXAMPLE A Marathon Runner s Distance A marathon runner is running a mile marathon. B P.M., the runner has run 3 miles. B P.M., the runner has run 15 miles, as shown in Figure B.. Find the average rate of change of the runner and use this rate of change to find the equation that relates the runner s distance to the time. Use the model to estimate the time when the runner will finish the marathon. P.M. P.M. 3 mi. 15 mi. Not drawn to scale A marathon is approimatel miles long. FIGURE B. Let represent the runner s distance and let t represent the time. Then the two points that represent the runner s positions are t 1, 1, 3 and t,, 15. So, the average rate of change of the runner is Average rate of change 1 t t 1 So, an equation that relates the runner s distance to the time is 1 m t t 1 Point-slope form 3 t Substitute 1 3, t 1, and m. t 9. Linear model To find the time when the runner will finish the marathon, let and solve for t to obtain t 9 35 t 5.8 t. 15 3 miles per hour. So, continuing at the same rate, the runner will finish the marathon at about P.M.
APPENDIX B Mathematical Modeling B5 0,000 0,000 (0,,000) EXAMPLE 5 Population of Anchorage, Alaska Between 1980 and 1998, the population of the cit of Anchorage, Alaska, increased at an average rate of approimatel 500 people per ear. In 1980, the population was 17,000. Find a mathematical model that gives the population of Anchorage in terms of the ear, and use the model to estimate the population in 000. (Source: U.S. Census Bureau) Let represent the population of Anchorage, and let t represent the calendar ear, with t 0 corresponding to 1980. Letting t 0 correspond to 1980 is convenient because ou were given the population in 1980. Now, using the rate of change of 500 people per ear, ou have Population 0,000 00,000 180,000 10,000 FIGURE B.5 (0, 17,000) 8 10 1 1 1 18 0 Year (0 1980) t Rate of change mt b 1980 population 500t 17,000. Using this model, ou can estimate the 000 population to be 000 population 500 0 17,000,000. The graph is shown in Figure B.5. (In this particular eample, the linear model is quite good the actual population of Anchorage, Alaska, in 000 was 0,000.) In Eample 5, note that in the linear model the population changed b the same amount each ear [see Figure B.(a)]. If the population had changed b the same percent each ear, the model would have been eponential, not linear [see Figure B.(b)]. (You will stud eponential models in Appendi B.3.) 35 30 (5, 3) 35 30 (5, 3) 5 5 0 15 10 5 =. t+ 1 (0, 1) 1 3 5 t 0 15 10 5 = (0, 1) 1 3 5 (a) Linear model changes b same amount each ear. FIGURE B. (b) Eponential model changes b same percent each ear.
B APPENDIX B Mathematical Modeling Scatter Plots Another tpe of linear modeling is a graphical approach that is commonl used in statistics. To find a mathematical model that approimates a set of actual data points, plot the points on a rectangular coordinate sstem. This collection of points is called a scatter plot. Once the points have been plotted, tr to find the line that most closel represents the plotted points. (In this section, we will rel on a visual technique for fitting a line to a set of points. If ou take a course in statistics, ou will encounter regression analsis formulas that can fit a line to a set of points.) EXAMPLE Fitting a Line to a Set of Points The scatter plot in Figure B.7(a) shows 35 different points in the plane. Find the equation of a line that approimatel fits these points. 3 1 3 1 1 3 5 1 3 5 (a) (b) FIGURE B.7 From Figure B.7(a), ou can see that there is no line that eactl fits the given points. The points, however, do appear to resemble a linear pattern. Figure B.7(b) shows a line that appears to best describe the given points. (Notice that about as man points lie above the line as below it.) From this figure, ou can see that the 1 best-fitting line has a -intercept at about 0, 1 and has a slope of about. So, the equation of the line is 1 1. If ou had been given the coordinates of the 35 points, ou could have checked the accurac of this model b constructing a table that compared the actual -values with the -values given b the model.
APPENDIX B Mathematical Modeling B7 EXAMPLE 7 Prize Mone at the Indianapolis 500 The total prize mone p (in millions of dollars) awarded at the Indianapolis 500 race from 1993 through 001 is shown in the table. Construct a scatter plot that represents the data and find a linear model that approimates the data. (Source: Indianapolis Motor Speedwa Hall of Fame) Year 1993 199 1995 199 1997 1998 1999 000 001 p $7.8 $7.8 $8.0 $8.11 $8.1 $8.7 $9.05 $9.8 $9. Prize mone (in millions of dollars) p 10 9 8 7 3 5 7 Year (3 1993) 8 9 10 11 t Let t 3 represent 1993. The scatter plot for the points is shown in Figure B.8. From the scatter plot, draw a line that approimates the data. Then, to find the equation of the line, approimate two points on the line: 5, 8 and 9, 9. The slope of this line is m p p 1 t t 1 9 8 9 5 1 FIGURE B.8 0.5 Using the point-slope form, ou can determine that the equation of the line is p 8 0.5 t 5 p 0.5t.75 Point-slope form Linear model To check this model, compare the actual p-values with the p-values given b the model (these are labeled p* below). t 3 5 7 8 9 10 11 p $7.8 $7.8 $8.0 $8.11 $8.1 $8.7 $9.05 $9.8 $9. p* $7.5 $7.75 $8.0 $8.5 $8.5 $8.75 $9.0 $9.5 $9.5
B8 APPENDIX B Mathematical Modeling B.1 Eercises 1. Falling Object In an eperiment, students measured the speed s (in meters per second) of a falling object t seconds after it was released. The results are shown in the table. t 0 1 3 s 0 11.0 19. 9. 39. A model for the data is s 9.7t 0.. (a) Plot the data and graph the model on the same set of coordinate aes. (b) Create a table showing the given data and the approimations given b the model. (c) Use the model to predict the speed of the object after falling 5 seconds. (d) Interpret the slope in the contet of the problem.. Cable TV The average monthl basic rate R (in dollars) for cable TV for the ears 199 through 1999 in the United States is given in the table. (Source: Paul Kagan Associates, Inc.) Year 199 1995 199 R 1. 3.07.1 Year 1997 1998 1999 R.8 7.81 8.9 A model for the data is R 1.508t 15.58, where t is the time in ears, with t corresponding to 199. (a) Plot the data and graph the model on the same set of coordinate aes. (b) Create a table showing the given data and the approimations given b the model. (c) Use the model to predict the average monthl basic rate for cable TV for the ear 005. (d) Interpret the slope in the contet of the problem. 3. Propert Ta The propert ta in a township is directl proportional to the assessed value of the propert. The ta on propert with an assessed value of $17,07 is $107. (a) Find a mathematical model that gives the ta T in terms of the assessed value v. (b) Use the model to find the ta on propert with an assessed value of $11,500. (c) Determine the ta rate.. Revenue The total revenue R is directl proportional to the number of units sold. When 5 units are sold, the revenue is $5. (a) Find a mathematical model that gives the revenue R in terms of the number of units sold. (b) Use the model to find the revenue when 3 units are sold. (c) Determine the price per unit. 5. The English and Metric Sstems The label on a roll of tape gives the amount of tape in inches and centimeters. These amounts are 500 inches and 170 centimeters. (a) Use the information on the label to find a mathematical model that relates inches to centimeters. (b) Use part (a) to convert 15 inches to centimeters. (c) Use part (a) to convert 50 centimeters to inches. (d) Use a graphing utilit to graph the model in part (a). Use the graph to confirm the results in parts (b) and (c).. The English and Metric Sstems The label on a bottle of soft drink gives the amount in liters and fluid ounces. These amounts are liters and 7.3 fluid ounces. (a) Use the information on the label to find a mathematical model that relates liters to fluid ounces. (b) Use part (a) to convert 7 liters to fluid ounces. (c) Use part (a) to convert 3 fluid ounces to liters. (d) Use a graphing utilit to graph the model in part (a). Use the graph to confirm the results in parts (b) and (c). The smbol indicates an eercise in which ou are instructed to use a calculator or graphing utilit.
APPENDIX B Mathematical Modeling B9 Civilian Labor Force In Eercises 7 and 8, use the graph, which shows the total civilian labor force N (in millions) in the United States from 1988 through 1999. (Source: U.S. Bureau of Labor Statistics) Total civilian labor force (in millions) 7. Using the data for 1988 through 1999, write a linear model for the total civilian labor force, letting t 8 represent 1988. Use the model to predict N in 00. Use a graphing utilit to graph the model and confirm the result. 8. In 199, 8 million of the labor force was unemploed. Approimate the percent of the labor force that was unemploed in 199. In Eercises 9 1, a scatter plot is shown. Determine whether the data appears linear. If so, determine the sign of the slope of a best-fitting line. 9. 10. 8 11. 1. 8 1 10 138 13 13 13 130 18 1 1 1 10 N (8, 11.7) 8 8 Year (8 1988) 8 8 (19, 139.) 8 9 10 11 1 13 1 15 1 17 18 19 8 8 t Rate of Change In Eercises 13 1, ou are given the dollar value of a product in 00 and the rate at which the value of the product is epected to change during the net 5 ears. Use this information to write a linear equation that gives the dollar value V of the product in terms of the ear t. Use a graphing utilit to graph the function. (Let t represent 00.) 00 Value Rate 13. $50 $15 increase per ear 1. $15 $.50 increase per ear 15. $0,00 $000 increase per ear 1. $5,000 $500 increase per ear Think About It In Eercises 17 0, match the description with its graph. Determine the slope and interpret its meaning in the contet of the problem. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) 0 30 0 10 18 1 8 8 17. A person is paing $10 per week to a friend to repa a $100 loan. 18. An emploee is paid $1.50 per hour plus $1.50 for each unit produced per hour. 19. A sales representative receives $0 per da for food plus $0.5 for each mile traveled. 0. A word processor that was purchased for $00 depreciates $100 per ear. (b) (d) 00 150 100 50 800 00 00 00 8 10 8
B10 APPENDIX B Mathematical Modeling 1. Investigation An instructor gives 0-point quizzes and 100-point tests in a mathematics course. The average quiz and test scores for si students given as ordered pairs,, where is the average quiz score and is the average test score, are 18, 87, 10, 55, 19, 9, 1, 79, 13, 7, and 15, 8. (a) Plot the points. (b) Use a ruler to sketch the best-fitting line through the points. (c) Find an equation for the line sketched in part (b). (d) Use part (c) to estimate the average test score for a person with an average quiz score of 17. (e) Describe the changes in parts (a) through (d) that would result if the instructor added points to each average test score.. Holders of Mortgage Debts The table shows the amount of mortgage debt (in billions of dollars) held b savings institutions and commercial banks for the ears 1995 through 1999 in the United States. (Source: The Federal Reserve Bulletin) Year 1995 199 1997 1998 1999 597 8 3 9 1090 115 15 1337 19 (a) Plot the points. (b) Use a ruler to sketch the best-fitting line through the points. (c) Find an equation for the line sketched in part (b). (d) Interpret the slope in the contet of the problem. 3. Advertising and Sales The table shows the advertising ependitures and sales volume for a compan for si randoml selected months. Both are measured in thousands of dollars. Month 1 3 5. 1..0. 1. 1. 0 18 0 0 180 1 (a) Plot the points. (b) Use a ruler to sketch the best-fitting line through the points. (c) Find an equation for the line sketched in part (b). (d) Interpret the slope in the contet of the problem. In Eercises 7, use a ruler to sketch the bestfitting line through the set of points, and find an equation of the line.. 5.. 7. (5, ) 8. Finding a Pattern Complete the table. The entries in the third row are the differences between consecutive entries in the second row. Describe the third row s pattern. (a) 0 1 3 5 (b) 9. Finding a Pattern Find m and b such that the equation m b ields the table. What does m represent? What does b represent? (a) 0 1 3 5 (b) ( 1, 1) (0, ) (, 3) (0, ) ( 3, 0) (1, 1) m b (3, ) (, ) 5 3 Differences 7 Differences m b (0, 7) (, ) (0, ) (, 1) (, 5) (, 3) (3, ) (, 0) 3 7 11 15 19 3 1 0 1 3 5 0 1 3 5 7 13 19 5 31