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Vocabulary

Review

Chapter 7 218

7-7 Exponential Growth and Decay

1. Match each situation in Column A with an equation that models it in Column B.

Column A Column B

A person begins with $100 and earns $2 each day. y 5 100 ? 2x

A person begins with $2 and earns $100 each day. y 5 100 1 2x

A person begins with $100. Each day the money doubles. y 5 2 1 100x

Vocabulary Builder

compound interest (noun) KAHM pownd IN trist

Related Words: principal (noun), simple interest (noun), interest rate (noun)

Definition: Compound interest is interest earned on both the principal and on any interest the account has already earned that remains in the account.

Use Your Vocabulary

Choose the correct word from the list to complete each sentence.

compound interest simple interest principal

2. The amount of money deposited in an account is the 9.

3. The amount of money earned on an investment only is 9.

4. Interest earned on both the principal and the interest left in the account is 9.

5. Use the situation below. Identify the principal, interest rate, time in years, and balance.

Suzanne deposits $1200 into a bank account paying 1.25% interest compounded annually for 5 years, at which point the account holds $1276.90.

principal: interest rate:

time in years: balance:

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d.Key Concept Exponential Growth

Problem 1

FPO

y

(0, a) xO

y a bx

b 1

y

(0, a)

xO

y a bx

0 b 1

219 Lesson 7-7

Modeling Exponential Growth

Got It? Suppose that in 1985, there were 285 cell phone subscribers in a small town. The number of subscribers increased by 75% each year after 1985. How many cell phone subscribers were in the small town in 1994? Write an expression to represent the equivalent monthly cell phone subscription increase.

8. Use the words exponent, growth factor, and initial amount to complete the diagram. Then find the value of the expression to the nearest whole number.

HSM11_A1MC_0707_T91262

9285 ∙ 1.75 value ofthe expression

new population

≈

9. The number of cell phone subscribers in 1994 will be about .

Key Concept Exponential Decay

6. Circle the numbers that could be growth factors.

85% 3 1.05 0.93

7. Circle the numbers that could be decay factors.

85% 3 1.05 0.93

The function y 5 a ? bx, where a . 0 and 0 , b , 1, models exponential decay.

The base b is the decay factor.

Algebra

HSM11_A1MC_0707_T91264

y = a • b x

initial amount (when x = 0)

�e base, which is lessthan 1, is the decay factor.

exponent

The function y 5 a ? bx, where a . 0 and b . 1, models exponential growth.

The base b is the growth factor.

Algebra

HSM11_A1MC_0707_T91261

y = a • b x

initial amount (when x = 0)

�e base, which is greaterthan 1, is the growth factor.

exponent

12

12

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Chapter 7 220

Problem 2

10. Fill in each blank to make a true statement.

In one year there are months. In x years there are months.

The yearly growth factor of cell phone subscribers is .

1.75 represents the yearly increase of cell phone subscriptions where x is the number of years.

11. Circle the expression that you could use to find the monthly increase in cell phone use in one year.

1.7512x 1.751

12 1.7512x

12. Let m represent the number of months. Write an expression that represents the equivalent monthly cell phone subscription increase.

Compound Interest

Got It? Suppose that when your friend was born, your friend’s parents deposited $2000 in an account paying 4.5% interest compounded monthly. What will the account balance be after 18 yr?

13. Draw a line from each variable in Column A to its value from the problem in Column B.

Column A Column B

P 18

r 0.045

n 2000

t 12

14. Substitute the values into the formula A 5 PQ1 1 rnR

nt to find the account balance.

15. The balance in the account after 18 yr will be $ .

The formula below gives the balance, A, in an account that earns compound interest. The formula is an exponential function with initial amount P and growth

factor Q1 1 rnR

n.

HSM11_A1MC_0707_T91263

A = P ∙1 + ∙nt

Principal(initial deposit)

number ofcompounding periodsper year

annual interest rate(as a decimal)

time in yearsrn

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Math Success

Now Iget it!

Need toreview

0 2 4 6 8 10

Lesson Check

Problem 3

221 Lesson 7-7

Check off the vocabulary words that you understand.

exponential growth compound interest exponential decay

Rate how well you can model exponential growth and decay.

Modeling Exponential Decay

Got It? The kilopascal is a unit of measure for atmospheric pressure. The atmospheric pressure at sea level is about 101 kilopascals. For every 1000-m increase in altitude, the pressure decreases about 11.5%. What is the pressure at an altitude of 5000 m?

16. Between sea level and 5000 m, the atmospheric pressure will decrease

5000 ÷ , or times.

17. Write and simplify an equation to find the atmospheric pressure at 5000 m.

y 101 (1 )

initial amount(of pressure)

decay factor(1 minus the percent decrease written as a decimal)

exponent(number of times the pressure decreases)

18. To the nearest whole unit, the pressure at 5000 m is about kilopascals.

• Do you UNDERSTAND?

Reasoning How can you simplify the compound interest formula when the interest is compounded annually? Explain.

19. When interest is compounded annually, the value of n in Pa1 1rnbnt

is .

20. Now simplify the compound interest formula.

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