Lesson 8-8 Exponential Growth and Decay 437

Exponential Growth and DecayLesson Preview

Part 1 Exponential Growth

In 1990, Floridas population was about 13 million. Since 1990, the statespopulation has grown about 1.7% each year. This means that Floridas populationis growing exponentially.

To find Floridas population in 1991, multiply the 1990 population by 1.7% and addthis to the 1990 population. So the population in 1991 is (1.7% + 100%) of the1990 population, or 101.7% of the 1990 population. Here is a function that modelsFloridas population since 1990.

population in millions

y = 13.0(1.017)x number of years since 1990

101.7% as a decimal

The following is a general rule for modeling exponential growth.

8-88-8

Check Skills Youll Need (For help, go to Lesson 4-3.)

Use the formula I prt to find the interest for principal p, interest rate r, andtime t in years.

1. principal: $1000; interest rate: 5%; time: 2 years $100

2. principal: $360; interest rate: 6%; time: 3 years $64.80

3. principal: $2500; interest rate: 4.5%; time: 2 years $225

4. principal: $1680; interest rate: 5.25%; time: 4 years $352.80

5. principal: $1350; interest rate: 4.8%; time: 5 years $324

New Vocabulary exponential growth growth factor compound interest interest period exponential decay decay factor

What Youll LearnTo model exponentialgrowth

To model exponentialdecay

. . . And WhyTo find the balance of a bank account, as in Examples 2 and 3

OBJECTIVE

2

OBJECTIVE

1

1OBJECTIVE

1Interactive lesson includes instant self-check, tutorials, and activities.

Exponential Growth

Key Concepts Rule Exponential Growth

can be modeled with the function

y = a ? bx for a . 0 and b . 1.

starting amount (when x = 0)

y = a ? bx exponent

The base, which is greater than 1, is the growth factor.

Exponential growth

ConnectionReal-World

In 2000, Floridas populationwas about 16 million. Roughly23% of the population wasunder the age of 18.

1. Plan

Lesson Preview

Check Skills Youll Need

Proportions and Percent EquationsLesson 4-3Exercise 53Extra Practice, p. 705

Lesson Resources

Teaching ResourcesPractice, Reteaching, Enrichment

Reaching All StudentsPractice Workbook 8-8Spanish Practice Workbook 8-8Technology Activities 8Hands-On Activities 19Basic Algebra Planning Guide 8-8

Presentation Assistant Plus!Transparencies Check Skills Youll Need 8-8 Additional Examples 8-8 Student Edition Answers 8-8 Lesson Quiz 8-8PH Presentation Pro CD 8-8

Computer Test Generator CD

TechnologyResource Pro CD-ROM Computer Test Generator CDPrentice Hall Presentation Pro CD

www.PHSchool.comStudent Site Teacher Web Code: aek-5500 Self-grading Lesson QuizTeacher Center Lesson Planner Resources

Plus

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437

Before the LessonDiagnose prerequisite skills using: Check Skills Youll Need

During the LessonMonitor progress using: Check Understanding Additional Examples Standardized Test Prep

After the LessonAssess knowledge using: Lesson Quiz Computer Test Generator CD

Ongoing Assessment and Intervention

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http://www.phschool.com/atschool/phmath/aek-5500.html

Modeling Exponential Growth

Medical Care Since 1985, the daily cost of patient care in community hospitals inthe United States has increased about 8.1% per year. In 1985, such hospital costswere an average of $460 per day.

a. Write an equation to model the cost of hospital care.

Relate y = a ? bx Use an exponential function.

Define Let x = the number of years since 1985.Let y = the cost of community hospital care at various times.Let a = the initial cost in 1985, $460.Let b = the growth factor, which is 100% + 8.1% = 108.1% = 1.081.

Write y = 460 ? 1.081x

b. Use your equation to find the approximate cost per day in 2000.

y = 460 ? 1.081x

y = 460 ? 1.08115 2000 is 15 years after 1985, so substitute 15 for x.

< 1480 Use a calculator. Round to the nearest dollar.

The average cost per day in 2000 was about $1480.

a. Suppose your community has 4512 students this year. The student population isgrowing 2.5% each year. Write an equation to model the student population.

b. What will the student population be in 3 years?

When a bank pays interest on both the principal and the interest an account hasalready earned, the bank is paying An is thelength of time over which interest is calculated.

Compound Interest

Savings Suppose your parents deposited $1500 in an account paying 6.5% interestcompounded annually (once a year) when you were born. Find the account balanceafter 18 years.

Relate y = a ? bx Use an exponential function.

Define Let x = the number of interest periods.Let y = the balance.Let a = the initial deposit, $1500.Let b = 100% + 6.5% = 106.5% = 1.065.

Write y = 1500 ? 1.065x

= 1500 ? 1.06518 Once a year for 18 years is 18 interest periods.Substitute 18 for x.

< 4659.98 Use a calculator. Round to the nearest cent.

The balance after 18 years will be $4659.98.

a. Suppose the interest rate on the account in Example 2 was 8%. How muchwould be in the account after 18 years? $5994.03

b. Another formula for compound interest is B = p(1 + r)x, where B is thebalance, p is the principal, and r is the interest rate in decimal form. Use thisformula to find the balance in the account in part (a). $5994.03

c. Critical Thinking Explain why the two formulas for finding compound interestare actually the same.

Check Understanding 22

EXAMPLEEXAMPLE22

interest periodcompound interest.

Check Understanding 11

EXAMPLEEXAMPLE11

Calculator Hint

To evaluate 460 ? 108115, press

460 1.081

15 .

438 Chapter 8 Exponents and Exponential Functions

Deposit $1500

Interest compounded annually 6.5%

Balance after 18 years $4659.98

a. y 4512 ?1.025x b. about 4859 students

(1 r) is the same as 100% 100r% written as a decimal.

438

2. Teach

Math Background

Exponential functions are widelyused to model many types ofgrowth and decay. The graph ofan exponential growth functionrises from left to right at an ever-increasing rate while that of anexponential decay function fallsfrom left to right at an ever-decreasing rate.

Teaching Notes

Teaching Tip

Even though students mayunderstand the word exponent,they may not understand whatgrowing exponentially means.Have students extend this table.

Multiply by 2 Square2 24 48 16

64 256

Continue until the student sees that the geometric sequenceformed with the common ratio 2grows much more slowly than thesequence formed by squaring(using the exponent 2).

Alternative Method

Have students solve the problemusing the [TABLE] function on agraphing calculator. First put theequation into . Then press2nd [TABLE]. Use the arrows toscroll to x = 18. The amount inthe y-column is 4660. Theamounts in the y-column havebeen rounded to the nearesttenth. Ask students to find how long it took to double the amount deposited. Guidestudents to look in the y-column for the amount closest to 3000. a little over 11 years

Y=

EXAMPLEEXAMPLE22

EXAMPLEEXAMPLE11

OBJECTIVE

1

Reaching All StudentsBelow Level Have students draw a treediagram illustrating the following: oneperson sends an e-mail to two friends;then each person forwards the e-mailto two friends, and so on.

Advanced Learners Ask students toexplain whether the consumption perperson of whole milk in the UnitedStates as modeled in Example 5 willever reach 0 gal/person.

English LearnersSee note on page 440.Error PreventionSee note on page 441.

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When interest is compounded quarterly (four times per year), you divide theinterest rate by 4, the number of interest periods per year. To find the number ofpayment periods, you multiply the number of years by the number of interestperiods per year.

Compound Interest

Savings Suppose the account in Example 2 paid interest compounded quarterlyinstead of annually. Find the account balance after 18 years.

Relate y = a ? bx Use an exponential function.

Define Let x = the number of interest periods.Let y = the balance.Let a = the initial deposit, $1500.

Let b = 100% + There are 4 interest periods in 1 year,so divide the interest into 4 parts.

= 1 + 0.01625 = 1.01625

Write y = 1500 ? 1.01625x

= 1500 ? 1.0162572Four interest periods a year for 18 years is 72 interest periods. Substitute 72 for x.

< 4787.75 Use a calculator. Round to the nearest cent.

The balance after 18 years will be $4787.75.

a. Suppose the account in Example 3 paid interest compounded monthly. Howmuch money would be in the account after 18 years? $4817.75

b. You deposit $200 into an account earning 5%, compounded monthly. How muchwill be in the account after 1 year? After 2 years? After 5 years?

Part 2 Exponential Decay

The graphs at the right show exponentialgrowth and exponential decay. For exponential growth, as x increases,y increases exponentially. For exponential decay, as x increases, y decreases exponentially.

O

y

4

x2 4

2

exponentialgrowth

y 2(1.5)x

6

6

8

exponentialdecay

y 2(0.5)x

Check Understanding 33

6.5%4

EXAMPLEEXAMPLE33

Annual Interest Rate of 8%

Compounded

annually

Periods per Year

1

Interest Rate per Period