74 CHAPTER 1 INTRODUCTION TO PROBLEM SOLVING AND MATHEMATICAL MODELS

g. I f 91 tickets are sold, use the equation mo'del developed in part f to determine the amount donated to the summer camp fund.

h. I f the maximum capacity of the rink is 200 people, what is the maximum amount that can be donated?

S U M M A R Y Activity 1.8

You will learn more about using graphs as mathematical models in Activity 1.13.

1. A mathematical model is a mathematical form, such as a formula, an equation, a graph, or a table, that fits or approximates the important features of a given situation.

2. Mathematical modeling is the process of developing a mathematical model for a given situation.

E X E R C I S E S Activity 1.8

1. The speed, s, of an ant (in centimeters per second) is related to the temperature, /* (in degrees Celsius), by the formula

s = 0 .167 / - 0.67.

a. I f the temperature is 40°C, how fast is the ant moving?

b. I f an ant is moving at 4 centimeters per second, what is the temperature?

2. The pressure, />, of water (in pounds per square foot) at a depth of d feet below the surface is given by the formula

a. I f a diver is 100 feet below the surface, what is the pressure of the water on the diver?

ACTIVITY 1.8 MATHEMATICAL MODELING 75

b. On November 14, 1993, Francisco Ferreras reached a record depth for breath-held diving. During the dive, he experienced a pressure of 201 pounds per square foot. What was his record depth?

3. The following formula is used by the National Football League (NFL) to calculate quarterback ratings:

250C+ 12.5 r+ lOOOr- 1250/+ 6.25^ 3A

where /? — quarterback rating A = passes attempted C = passes completed r = passing yardage T = touchdown passes / = number of interceptions

In the 2005-2006 regular season, Tom Brady, quarterback for the New England Patriots, and Brett Favre, for the Green Bay Packers, had the following player statistics:

Tom Brady

Brett Favre

PASSES

AXTEMf i JE f t i

530

607

PASSES

C O M P L E T E D ,

334

372

PASSING

YARDAGE

4110

3881

NUMBER O F

T O U C H D O W N

PASSES

26

20

NUMBER O F

I N T E R C E P T I O N S

14

29

a. Determine the quarterback rating for Tom Brady for the 2005-2006 NFL football season.

b. Determine the quarterback rating for Brett Favre.

c. Visit www.nfl.com and select stats to obtain the rating of your favorite quarterback.

4. a. You want to invest money in order to receive the best return. You have two options.

Option 1: Invest at 6% simple annual interest for 10 years.

Option 2: Invest at 5% interest compounded annually.

The following tables model the growth of $2000 over a 10-year period using the two options.

76 CHAPTER 1 INTRODUCTION TO PROBLEM SOLVING AND MATHEMATICAL MODELS

NUMBER O F Y E A R S

i I 2 I 3 I 4 I 5 I 6 I 7 I 8 I 9 I lo 6% SIMPLE

A N N U A L

INTEREST 2120 2240 2360 2480 2600 2720 2840 2960 3080 3200 5% C O M P O U N D E D

ANNUALLY 2100 2205 2315.30 2431 2552.60 2680.20 2814.20 2954.90 3102.70 3257.80

Describe any trends or patterns that you observe in the data.

b. The amount of your investment in option 1 can be determined by the following formula

A = F + Prt,

where A = amount of the investment P = principal or amount invested r = annual percentage rate (expressed as a decimal) / = number of years invested

Use the formula to determine the amount of your $2000 investment in option 1 after 20 years. What is the total amount of interest earned?

c. The amount of your investment in option 2 can be determined by

A^P{\+ r)\

where A = amount of the investment P = principal /• = annual percentage rate (expressed as a decimal) / = number of years invested

Use the formula to determine the amount of your $2000 investment in option 2 after 20 years.

d. Which option would you choose? Explain.

ACTIVITY 1.8 MATHEMATICAL MODELING 77

5. The following table gives the number of people infected by the flu over a given number of months.

NUMBER O F MONTHS 0 1 2 3 4 5

NUMBER O F P E O P L E I N F E C T E D 1 5 13 33 78 180

Describe any trends or patterns that you observe.

6. The value of almost everything you own (assets), such as a car, computer, or house, depreciates (goes down) over time. When an asset's value decreases by a fixed amount each year, the depreciation is called straight-line depreciation.

Suppose your truck has an initial value of $12,400 and depreciates $820 per year.

a. State a question that you might want answered in this situation.

b. What two variables are involved in this problem?

c. Which variable can best be designated as the dependent variable? As the indepen­dent variable?

d. Complete the following table.

I N D E P E N D E N T V A R I A B L E 1 2 3 4- 5

D E P E N D E N T V A R I A B L E

e. State in words the relationship between the independent and dependent variables.

f. Use appropriate letters to represent the variables involved and translate the written statement in part e as an equation.

g. I f you plan to keep the truck for 7 years, determine the value of the truck at the end of this period. Explain the process you used.

h. What assumption was made regarding the rate of depreciation of the tnick? Does this seem reasonable?