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

Solutions
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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

Solutions
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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

Solutions
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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

Solutions
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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

Solutions
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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

Solutions
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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

Solutions
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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
  [Add.]
Step: 8
So, the exponential growth model for the profit is y = 13333(1.04)8.
Correct Answer is :   y = 13333(1.04)8

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