#### Solved Examples and Worksheet for Writing Exponential Expressions - Growth and Decay

Q1The profit made by a software company in 1992 was $122 million. Identify an expression to represent the profit made by the company after n years, if the profit is growing exponentially at a rate of 0.3% per year. A. 122(1 + 1.003)n B. 122(1 - 0.003)n C. 122(0.003)n D. 122(1.003)n Step: 1 Let y be the profit made by the company after n years. Step: 2 Profit made by the company in 1992 was$122 million.
Step: 3
The exponential growth rate, r is 0.3%.
Step: 4
y = C(1 + r)t
[Write exponential growth model.]
Step: 5
= 122(1 + 0.003)n
[Substitute C = 122, r = 0.003 and t = n.]
Step: 6
= 122(1.003)n
[Simplify.]
Step: 7
So, the profit made by the company after n years will be $122(1.003)n million. Correct Answer is : 122(1.003)n Q2A businessman made a profit of$12,143 in 1990. The profit increased by 2% per year for the next 7 years. Identify the equation that represents his profit.
A. y = 12143(1.02)7
B. y = 12143(2)6
C. y = 1.02(12143)7
D. y = 12143(2)8

Step: 1
Let y be the profit.
Step: 2
The initial profit in the business is $12143. Step: 3 The rate of increase of profit, r is 2% or 0.02. Step: 4 The number of years, t is 7. Step: 5 y = C(1 + r)t [Write exponential growth model.] Step: 6 y =12143(1 + 0.02)7 [Substitute C = 12143, r = 0.02 and t = 7.] Step: 7 y = 12143(1.02)7 [Add.] Step: 8 So, the equation for the profit is y = 12143(1.02)7. Correct Answer is : y = 12143(1.02)7 Q3Brad bought a bike for$4,300. The bike's value decreases by 13% each year. Identify an exponential decay model to represent the situation.

A. y = 4300(0.87)t
B. y = 4300(0.13)t
C. y = 4300 - 13%t
D. y = 4300(14)t

Step: 1
Let y be the value of the bike.
Step: 2
Let t be the number of years of ownership.
Step: 3
The initial value of the bike C is $4300. Step: 4 The decay rate r is 13% or 0.13. Step: 5 y = C(1 - r)t [Write exponential decay model.] Step: 6 = 4300(1 - 0.13)t [Replace C with 4300 and r with 0.13.] Step: 7 = 4300(0.87)t [Subtract 0.13 from 1.] Step: 8 The exponential decay model is y = 4300(0.87)t. Correct Answer is : y = 4300(0.87)t Q4A company purchased machinery in the year 1990 for$29,000. Its cost depreciates at a rate of 3% per year. Identify an exponential decay model to represent the cost of the machinery.
A. y = 29100(0.97)t
B. y = 28900(0.97)t
C. y = 29000(0.97)t
D. y = 29000(0.87)t

Step: 1
The initial value of the machinery, C is $29000. Step: 2 The decay rate, r is 3% = 0.03. Step: 3 The exponential decay can be modeled by the equation y = C(1 - r)t. Step: 4 y = 29000(1 - 0.03)t [Replace C with 29000, and r with 0.03.] Step: 5 y = 29000(0.97)t [Subtract 0.03 from 1.] Step: 6 The exponential decay model that represents the depreciation of the machinery is y = 29000(0.97)t. Correct Answer is : y = 29000(0.97)t Q5Victor purchased a refrigerator for$4,000 in the year 2000. Its value depreciates by 5% each year. Identify an exponential decay model to represent this situation.
A. y = 3800(0.98)t
B. y = 4000(1.05)t
C. y = 4000(0.85)t
D. y = 4000(0.95)t

Step: 1
The initial value of the refrigerator C is $4000. Step: 2 The decay rate r is 5% or 0.05. Step: 3 The exponential decay can be modeled by the equation y = C(1 - r)t. Step: 4 y = 4000(1 - 0.05)t [Replace C with 4000, and r with 0.05.] Step: 5 y = 4000(0.95)t [Subtract.] Step: 6 The exponential decay model that represents the situation is y = 4000(0.95)t. Correct Answer is : y = 4000(0.95)t Q6The average length of a person's hair at birth is 0.23 inches. The length of the hair increases by about 10% each day during the first six weeks. Choose the model that represents the average length of the hair during the first six weeks. A. y = 0.23(1.1)t B. y = - 0.23(1.1)t C. y = 0.23(0.1)t D. y = 1.1(0.23)t Step: 1 Let y be the length of the hair during the first six weeks and t be the number of days. Step: 2 y = C(1 + r)t [Write exponential growth model.] Step: 3 = 0.23(1 + 0.1)t [Replace C = 0.23 and r = 10% = 0.1.] Step: 4 = 0.23(1.1)t [Add.] Step: 5 The model for the length of the hair in first six weeks is y = 0.23(1.1)t. Correct Answer is : y = 0.23(1.1)t Q7A business man made a profit of$13333 in 1990. The profit increased by 4% per year for the next 8 years. Identify an exponential growth model for the profit.

A. y = 13333(4)9
B. y = 1.04(13333)8
C. y = 13333(4)7
D. y = 13333(1.04)8

Step: 1
Let y be the profit.
Step: 2
The initial profit in the business, C is \$13333.
Step: 3
The rate of increase of profit, r is 4% or 0.04.
Step: 4
The number of years, t is 8.
Step: 5
y = C(1 + r)t
[Write exponential growth model.]
Step: 6
y =13333(1 + 0.04)8
[Substitute C = 13333, r = 0.04 and t = 8.]
Step: 7
y = 13333(1.04)8