appendixb mathematical modeling b1 appendix b mathematical...

APPENDIX B Mathematical Modeling B1

B.1 Modeling Data with Linear Functions

Introduction • Direct Variation • Rates of Change •Scatter Plots

IntroductionThe primary objective of applied mathematics is to find equations or mathemat-ical models that describe real-world phenomena. In developing a mathematicalmodel to represent actual data, you should strive for two (often conflicting)goals—accuracy and simplicity. That is, you want the model to be simple enoughto be workable, yet accurate enough to produce meaningful results.

EXAMPLE 1 A Mathematical Model

The total annual amounts of advertising expenses y (in billions of dollars) in theUnited States from 1990 through 1999 are shown in the table. (Source: McCannErickson)

A linear model that approximates this data is

where y represents the advertising expenses (in billions of dollars) and t repre-sents the year, with corresponding to 1990. Plot the actual data and themodel on the same graph. How closely does the model represent the data?

Solution

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. Youcan see how well the model fits by comparing the actual values of y with thevalues of y given by the model (these are labeled y* in the table below).

t � 0

0 ≤ t ≤ 9y � 10.19t � 116.5,

Year 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999

y 129.6 127.5 132.7 139.5 151.7 162.9 175.2 187.5 201.6 215.2

t 0 1 2 3 4 5 6 7 8 9

y 129.6 127.5 132.7 139.5 151.7 162.9 175.2 187.5 201.6 215.2

y* 116.5 126.7 136.9 147.1 157.3 167.5 177.6 187.8 198.0 208.2

Year (0 1990)↔

eA

dver

tisin

gxp

ense

s (i

n bi

llion

s of

dol

lars

)

1 2 3 4 5 6 7 8 9

20

40

60

80

100

120

140

160

180

200

220

t

y

y = 10.19t +116.5

FIGURE B.1

Appendix B Mathematical Modeling

B2 APPENDIX B Mathematical Modeling

Direct VariationThere are two basic types of linear models. The more general model has a y-intercept that is nonzero: where The simpler one,has a y-intercept that is zero. In the simpler model, y is said to vary directly as x,or to be proportional to x.

EXAMPLE 2 State Income Tax

In Pennsylvania, the state income tax is directly proportional to gross income.Suppose you were working in Pennsylvania and your state income tax deductionwas $42 for a gross monthly income of $1500.00. Find a mathematical model thatgives the Pennsylvania state income tax in terms of the gross income.

Solution

Let y represent the state income tax in dollars and x represent the gross income indollars. Then you know that y and x are related by the equation

You are given when By substituting these values into the equa-tion you can find the value of m.

Direct variation model

Substitute and

Divide each side by 1500.

So, the equation (or model) for state income tax in Pennsylvania is In other words, Pennsylvania has a state income tax rate of 2.8% of the grossincome. The graph of this equation is shown in Figure B.2.

y � 0.028x.

0.028 � m

x � 1500.y � 42 42 � m�1500�

y � mx

y � mx,x � 1500.y � 42

y � mx.

y � mx,b � 0.y � mx � b,

Direct Variation

The following statements are equivalent.

1. y varies directly as x.

2. y is directly proportional to x.

3. for some nonzero constant m.

m is the constant of variation or the constant of proportionality.

y � mx

Gross income (in dollars)

Stat

e in

com

e ta

x (i

n do

llars

)

x1000 2000 3000 4000

20

40

60

80

100

(1500, 42)

y

FIGURE B.2

Most measurements in the English system and the metric system are directlyproportional. The next example shows how to use a direct proportion to convertbetween miles per hour and kilometers per hour.

EXAMPLE 3 The English and Metric Systems

You are traveling at a rate of 64 miles per hour. You switch your speedometerreading to metric units and notice that the speed is 103 kilometers per hour. Usethis information to find a mathematical model that relates miles per hour to kilo-meters per hour.

Solution

Let y represent the speed in miles per hour and x represent the speed in kilometersper hour. Then you know that y and x are related by the equation

You are given when By substituting these values into theequation you can find the value of m.

Direct variation model

Substitute and

Divide each side by 103.

Use a calculator.

So, the conversion factor from kilometers per hour to miles per hour is approxi-mately 0.62136, and the model is

The graph of this equation is shown in Figure B.3.

Once you have found a model that converts speeds from kilometers per hourto miles per hour, you can use the model to convert other speeds from the metricsystem to the English system, as shown in the table.

NOTE The conversion equation can be approximated by thesimpler equation For instance, to convert 40 kilometers per hour,divide by 8 and multiply by 5 to obtain 25 miles per hour.

y �58 x.

y � 0.62136x

y � 0.62136x.

0.62136 � m

64103

� m

x � 103.y � 64 64 � m�103� y � mx

y � mx,x � 103.y � 64

y � mx.


Kilometers per hour 20.0 40.0 60.0 80.0 100.0 120.0

Miles per hour 12.4 24.9 37.3 49.7 62.1 74.6

Kilometers per hour

Mile

s pe

r ho

ur

x20 40 60 80 100 120

10

20

30

40

50

60

70

80(103, 64)

y

FIGURE B.3


Rates of ChangeA second common type of linear model is one that involves a known rate ofchange. In the linear equation

you know that m represents the slope of the line. In real-life problems, the slopecan often be interpreted as the rate of change of y with respect to x. Rates ofchange should always be listed in appropriate units of measure.

EXAMPLE 4 A Marathon Runner’s Distance

A marathon runner is running a 26 mile marathon. By 2 P.M., the runner has run3 miles. By 4 P.M., the runner has run 15 miles, as shown in Figure B.4. Find theaverage rate of change of the runner and use this rate of change to find the equa-tion that relates the runner’s distance to the time. Use the model to estimate thetime when the runner will finish the marathon.

Solution

Let y represent the runner’s distance and let t represent the time. Then the twopoints that represent the runner’s positions are

and

So, the average rate of change of the runner is

So, an equation that relates the runner’s distance to the time is

Point-slope form

Substitute and

Linear model

To find the time when the runner will finish the marathon, let and solvefor t to obtain

So, continuing at the same rate, the runner will finish the marathon at about 6 P.M.

5.8 � t.

35 � 6t

26 � 6t � 9

y � 26

y � 6t � 9.

m � 6.y1 � 3, t1 � 2, y � 3 � 6�t � 2� y � y1 � m�t � t1�

� 6 miles per hour.

�15 � 34 � 2

Average rate of change �y2 � y1

t2 � t1

�t2, y2� � �4, 15�.�t1, y1� � �2, 3�

y � mx � b

3 mi. 15 mi.Not drawn to scale

2 P.M. 4 P.M.

A marathon is approximately 26 miles long.FIGURE B.4

EXAMPLE 5 Population of Anchorage, Alaska

Between 1980 and 1998, the population of the city of Anchorage, Alaska,increased at an average rate of approximately 4500 people per year. In 1980, thepopulation was 174,000. Find a mathematical model that gives the population ofAnchorage in terms of the year, and use the model to estimate the population in2000. (Source: U.S. Census Bureau)

Solution

Let y represent the population of Anchorage, and let t represent the calendar year,with corresponding to 1980. Letting correspond to 1980 is con-venient because you were given the population in 1980. Now, using the rate ofchange of 4500 people per year, you have

Using this model, you can estimate the 2000 population to be

The graph is shown in Figure B.5. (In this particular example, the linear model isquite good—the actual population of Anchorage, Alaska, in 2000 was 260,000.)

In Example 5, note that in the linear model the population changed by thesame amount each year [see Figure B.6(a)]. If the population had changed by thesame percent each year, the model would have been exponential, not linear [seeFigure B.6(b)]. (You will study exponential models in Appendix B.3.)

(a) Linear model changes by (b) Exponential model changessame amount each year. by same percent each year.

FIGURE B.6

x

(0, 1)

(5, 32)

1 2 3 4 5 6

5

10

15

20

25

30

35

y = 2x

y

1

5

t

(5, 32)

(0, 1)

2 3 4 5 6

10

15

20

25

30

35

y t= 6.2 + 1

y

2000 population � 4500�20� � 174,000 � 264,000.

y � 4500t � 174,000.

mt � by �

t � 0t � 0


Rate ofchange

1980population

Year (0 1980)↔

Popu

latio

n

2 4 6 8 10 12 14 16 18 20

160,000

180,000

t

y

200,000

220,000

240,000

260,000

(0, 174,000)

(20, 264,000)

FIGURE B.5


Scatter PlotsAnother type of linear modeling is a graphical approach that is commonly usedin statistics. To find a mathematical model that approximates a set of actual datapoints, plot the points on a rectangular coordinate system. This collection ofpoints is called a scatter plot. Once the points have been plotted, try to find theline that most closely represents the plotted points. (In this section, we will relyon a visual technique for fitting a line to a set of points. If you take a course instatistics, you will encounter regression analysis formulas that can fit a line to aset of points.)

EXAMPLE 6 Fitting a Line to a Set of Points

The scatter plot in Figure B.7(a) shows 35 different points in the plane. Find theequation of a line that approximately fits these points.

(a) (b)

FIGURE B.7

Solution

From Figure B.7(a), you can see that there is no line that exactly fits the givenpoints. 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 asmany points lie above the line as below it.) From this figure, you can see that thebest-fitting line has a y-intercept at about and has a slope of about So, theequation of the line is

If you had been given the coordinates of the 35 points, you could have checkedthe accuracy of this model by constructing a table that compared the actual y-values with the y-values given by the model.

y �12

x � 1.

12 .�0, 1�

x1 2 3 4 5

1

2

3

4

y

x1 2 3 4 5

1

2

3

4

y

EXAMPLE 7 Prize Money at the Indianapolis 500

The total prize money p (in millions of dollars) awarded at the Indianapolis 500race from 1993 through 2001 is shown in the table. Construct a scatter plot thatrepresents the data and find a linear model that approximates the data. (Source:Indianapolis Motor Speedway Hall of Fame)

Solution

Let represent 1993. The scatter plot for the points is shown in Figure B.8.From the scatter plot, draw a line that approximates the data. Then, to find theequation of the line, approximate two points on the line: and Theslope of this line is

Using the point-slope form, you can determine that the equation of the line is

Point-slope form

Linear model

To check this model, compare the actual p-values with the p-values given by themodel (these are labeled p* below).

p � 0.25t � 6.75

p � 8 � 0.25�t � 5�

� 0.25

�14

�9 � 89 � 5

m �p2 � p1

t2 � t1

�9, 9�.�5, 8�

t � 3


Year 1993 1994 1995 1996 1997 1998 1999 2000 2001

p $7.68 $7.86 $8.06 $8.11 $8.61 $8.72 $9.05 $9.48 $9.62

t 3 4 5 6 7 8 9 10 11

p $7.68 $7.86 $8.06 $8.11 $8.61 $8.72 $9.05 $9.48 $9.62

p* $7.5 $7.75 $8.0 $8.25 $8.5 $8.75 $9.0 $9.25 $9.5

8 9 10 11

6

7

8

9

10

t3 4 5 6 7

p

Priz

e m

oney

(in

mill

ions

of

dolla

rs)

↔Year (3 1993)

FIGURE B.8


1. Falling Object In an experiment, students measuredthe speed s (in meters per second) of a falling object tseconds after it was released. The results are shown inthe table.

A model for the data is

(a) Plot the data and graph the model on the same setof coordinate axes.

(b) Create a table showing the given data and theapproximations given by the model.

(c) Use the model to predict the speed of the objectafter falling 5 seconds.

(d) Interpret the slope in the context of the problem.

2. Cable TV The average monthly basic rate R (in dol-lars) for cable TV for the years 1994 through 1999 inthe United States is given in the table. (Source: PaulKagan Associates, Inc.)

A model for the data is where tis the time in years, with corresponding to 1994.

(a) Plot the data and graph the model on the same setof coordinate axes.

(b) Create a table showing the given data and theapproximations given by the model.

(c) Use the model to predict the average monthlybasic rate for cable TV for the year 2005.


3. Property Tax The property tax in a township isdirectly proportional to the assessed value of the prop-erty. The tax on property with an assessed value of$17,072 is $1067.

(a) Find a mathematical model that gives the tax T interms of the assessed value v.

(b) Use the model to find the tax on property with anassessed value of $11,500.

(c) Determine the tax rate.

4. Revenue The total revenue R is directly proportion-al to the number of units sold x. When 25 units aresold, the revenue is $6225.

(a) Find a mathematical model that gives the revenueR in terms of the number of units sold x.

(b) Use the model to find the revenue when 32 unitsare sold.

(c) Determine the price per unit.

5. The English and Metric Systems The label on a rollof tape gives the amount of tape in inches and centimeters. These amounts are 500 inches and 1270centimeters.

(a) Use the information on the label to find a mathe-matical model that relates inches to centimeters.

(b) Use part (a) to convert 15 inches to centimeters.

(c) Use part (a) to convert 650 centimeters to inches.

(d) Use a graphing utility to graph the model in part(a). Use the graph to confirm the results in parts(b) and (c).

6. The English and Metric Systems The label on abottle of soft drink gives the amount in liters and fluidounces. These amounts are 2 liters and 67.63 fluidounces.

(a) Use the information on the label to find a mathe-matical model that relates liters to fluid ounces.

(b) Use part (a) to convert 27 liters to fluid ounces.

(c) Use part (a) to convert 32 fluid ounces to liters.

(d) Use a graphing utility to graph the model in part(a). Use the graph to confirm the results in parts(b) and (c).

t � 4R � 1.508t � 15.58,

s � 9.7t � 0.4.

B.1 Exercises

t 0 1 2 3 4

s 0 11.0 19.4 29.2 39.4

Year 1994 1995 1996

R 21.62 23.07 24.41

Year 1997 1998 1999

R 26.48 27.81 28.92

The symbol indicates an exercise in which you are instructed to use a calculator or graphing utility.


Civilian Labor Force In Exercises 7 and 8, use thegraph, which shows the total civilian labor force N(in millions) in the United States from 1988 through1999. (Source: U.S. Bureau of Labor Statistics)

7. Using the data for 1988 through 1999, write a linearmodel for the total civilian labor force, letting represent 1988. Use the model to predict N in 2004.Use a graphing utility to graph the model and con-firm the result.

8. In 1994, 8 million of the labor force was unem-ployed. Approximate the percent of the labor forcethat was unemployed in 1994.

In Exercises 9–12, 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.

11. 12.

Rate of Change In Exercises 13–16, you are giventhe dollar value of a product in 2002 and the rate atwhich the value of the product is expected tochange during the next 5 years. Use this informationto write a linear equation that gives the dollar valueV of the product in terms of the year t. Use a graph-ing utility to graph the function. (Let represent2002.)

2002 Value Rate

13. $2540 $125 increase per year

14. $156 $4.50 increase per year

15. $20,400 $2000 increase per year

16. $245,000 $5600 increase per year

Think About It In Exercises 17–20, match thedescription with its graph. Determine the slope andinterpret its meaning in the context of the problem.[The graphs are labeled (a), (b), (c), and (d).]

(a) (b)

(c) (d)

17. A person is paying $10 per week to a friend to repaya $100 loan.

18. An employee is paid $12.50 per hour plus $1.50 foreach unit produced per hour.

19. A sales representative receives $20 per day for foodplus $0.25 for each mile traveled.

20. A word processor that was purchased for $600depreciates $100 per year.

x

200

400

600

800

2 4 6 8

y

x2 4 6 8

24

18

12

6

y

x2 4 6 8 10

50

100

150

200

y

x

y

40

30

20

10

2 4 6 8

t � 2

2 4 6 8

2

4

6

8

x

y

x2 4 6 8

2

4

6

8

y

x2 4 6 8

2

4

6

8

y

x2 4 6 8

2

4

6

8

y

t � 8

Year (8 1988)↔

Tota

l civ

ilian

labo

r fo

rce

(in

mill

ions

)

(8, 121.7)

(19, 139.4)

158 9 10 11 12 13 14 16 17 18 19

120122

126

130

134

138

142

124

128

132

136

140

t

N


21. Investigation An instructor gives 20-point quizzesand 100-point tests in a mathematics course. Theaverage quiz and test scores for six students given asordered pairs where x is the average quiz scoreand y is the average test score, are

and

(a) Plot the points.

(b) Use a ruler to sketch the best-fitting line throughthe points.

(c) Find an equation for the line sketched in part (b).

(d) Use part (c) to estimate the average test score fora person with an average quiz score of 17.

(e) Describe the changes in parts (a) through (d) thatwould result if the instructor added 4 points toeach average test score.

22. Holders of Mortgage Debts The table shows theamount of mortgage debt (in billions of dollars) heldby savings institutions x and commercial banks y forthe years 1995 through 1999 in the United States.(Source: The Federal Reserve Bulletin)





23. Advertising and Sales The table shows the adver-tising expenditures x and sales volume y for acompany for six randomly selected months. Both aremeasured in thousands of dollars.





In Exercises 24–27, use a ruler to sketch the best-fitting line through the set of points, and find anequation of the line.

24. 25.

26. 27.

28. Finding a Pattern Complete the table. The entriesin the third row are the differences between consecu-tive entries in the second row. Describe the thirdrow’s pattern.

(a)

(b)

29. Finding a Pattern Find m and b such that the equa-tion yields the table. What does m rep-resent? What does b represent?

(a)

(b)

y � mx � b

x

(0, 7)

(4, 3)

(6, 0)−2 2 4

2

4

−2

(3, 2)

(2, 5)

y

x4

4

−2−2

6

(0, 2)

(1, 1)

(2, 2)

(3, 4)

(5, 6)

y

x2 4

4

2

−2−2

6

(2, 1)(0, 2)

( 2, 6)−

y

x

(4, 3)

(0, 2)( 1, 1)−

( 3, 0)− 2 4

4

6

y

�15, 82�.�13, 76�,�16, 79�,�19, 96�,�18, 87�, �10, 55�,

�x, y�,

Year 1995 1996 1997 1998 1999

x 597 628 632 644 669

y 1090 1145 1245 1337 1496

Month 1 2 3 4 5 6

x 2.4 1.6 2.0 2.6 1.4 1.6

y 202 184 220 240 180 164

0 1 2 3 4 5

3 7 11 15 19 23y � mx � b

x

0 1 2 3 4 5

�31�25�19�13�7�1y � mx � b

x

0 1 2 3 4 5

Differences

y � 5x � 3

x

0 1 2 3 4 5

Differences

y � �2x � 7

x

appendixb mathematical modeling b1 appendix b mathematical...

Documents