Solved Examples and Worksheet for Exponential Growth and Decay

Q1Which of the following equations represents exponential growth?

A. y = r(1 + C)
B. y = C(1 + r)t
C. y = Cr
D. y = r(1 + r)

Solutions
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Step: 1
Exponential growth can be modeled by the equation y = C (1 + r)t, where C is the initial amount, r is the growth rate and t is the time.
Correct Answer is :   y = C(1 + r)t
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
Q3Which of the following models is an exponential decay model?
A. y = 3(1.24)t
B. y = 8(0.67)t
C. y = 5t2
D. y = 2 + 5t

Solutions
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Step: 1
Exponential decay can be modeled by the equation y = C(1 - r)t, where C is the initial value, r is the decay rate where 0 < 1 - r < 1, and t, the time.
Step: 2
In the model y = 3(1.24)t, 1 - r = 1.24 and 1.24 > 1
Step: 3
The model y = 3(1.24)t is not an exponential decay model.
Step: 4
y = 2 + 5t and y = 5t2 are linear and quadratic models. So, they are not exponential models.
  [Compare with exponential decay model.]
Step: 5
In the model y = 8(0.67)t, 1 - r = 0.67 and 0 < 0.67 < 1
Step: 6
So, the model y = 8(0.67)t is an exponential decay model.
Correct Answer is :   y = 8(0.67)t
Q4Brad 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
Q5Arthur started a business in the year 1991. He got $47,000 profit in the first year. Each year his profit decreased by 2%. Identify an exponential decay model to represent his decreasing annual profit in the business.
A. y = 47000(0.98)t
B. y = 46999(0.98)t
C. y = 47000(0.88)t
D. y = 47001(1.08)t

Solutions
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Step: 1
Let y be Arthur's decreasing annual profit in the business.
Step: 2
The number of years that the profit decreased is t.
Step: 3
His initial profit C is $47000.
Step: 4
The decay rate r is 2% or 0.02.
Step: 5
The exponential decay can be modeled by the equation y = C(1 - r)t.
Step: 6
y = 47000(1 - 0.02)t
  [Replace C and r with the values $47000 and 0.02.]
Step: 7
y = 47000(0.98)t
  [Simplify.]
Step: 8
The exponential decay model of Arthur's decreasing annual profit in the business is y = 47000(0.98)t.
Correct Answer is :   y = 47000(0.98)t
Q6A 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
Q7Victor 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
Q8Which is the standard form of an exponential function?

A. y = abx, a is a constant ≠ 0, b ≠ 1,and x is a real number.
B. y = abx, a is a constant ≠ 0, b > 0, b ≠ 1,and x is a real number.
C. y = abx, a is a constant, b ≠ 1,and x is a real number.
D. None of the above.

Solutions
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Step: 1
The standard form of an exponential function is y = abx where a is a constant ≠ 0, b is the base, b > 0 and b ≠ 1, and x is a real number.
  [Definition.]
Correct Answer is :   y = abx, a is a constant ≠ 0, b > 0, b ≠ 1,and x is a real number.
Q9Which of the following models an exponential growth?

A. y = abx, a is a constant < 0, 0 < b < 1, and x is a real number
B. y = abx, a is a constant ≠ 0, 0 < b < 1, and x is a real number
C. y = abx, a is a constant > 0, b > 1, and x is a real number
D. None of the above

Solutions
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Step: 1
The exponential function y = abx, where a is a constant > 0, b > 1, and x is a real number models an exponential growth.
  [Definition.]
Correct Answer is :   y = abx, a is a constant > 0, b > 1, and x is a real number
Q10Which of the following models an exponential decay?
A. y = abx, a is a constant < 0, b > 1, and x is a real number
B. y = abx, a is a constant > 0, 0 < b < 1, and x is a real number
C. y = abx, a is a constant < 0, 0 < b < 1, and x is a real number
D. none of the above

Solutions
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Step: 1
The exponential function y = abx, where a is a constant > 0, 0 < b < 1, and x is a real number models an exponential decay.
  [Definition.]
Correct Answer is :   y = abx, a is a constant > 0, 0 < b < 1, and x is a real number
Q11Andy bought a bike for $5300. The bike's value decreases by 4% each year. Write an exponential decay model to represent the situation.
A. 5300(0.96)t
B. 5300(0.86)t
C. 5300(1.16)t
D. 5300(1.26)t
E. 5300(1.06)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 $5300.
Step: 4
The decay rate r is 4% or 0.04
Step: 5
y = C(1 - r)t
  
Step: 6
= 5300(1 - 0.04)t
  
Step: 7
= 5300(0.96)t
  
Step: 8
The exponential decay model is 5300(0.96)t.
Correct Answer is :   5300(0.96)t
Q12Which of the following is the exponential growth function whose initial value is 105 and whose base is 8?
A. y = 210(8x), x is a real number
B. y = 210x, x is a real number
C. y = 105(8x), x is a real number
D. y = 8(105x), x is a real number

Solutions
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Step: 1
y = a(bx), x is a real number be the required exponential growth function.
  [Standard exponential growth function.]
Step: 2
Here the base, b = 8
Step: 3
y = a(bx)
Step: 4
105 = a(8)0, a = 105
  [Replace y with 105, b with 8 and x with zero.]
Step: 5
So, the initial value of y is a = 105.
Step: 6
So, y = 105(8x) is the required expoential growth function.
Correct Answer is :   y = 105(8x), x is a real number
Q13The 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
Q14A company purchased machinery in 1990 for $29000. Its cost deprecates at a rate of 4% per year. Write an exponential decay model to represent the value of the machinery.
A. y = 28900(0.96)t
B. y = 29000(0.96)t
C. y = 29100(0.96)t
D. y = 29000(0.86)t

Solutions
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Step: 1
The initial value of the machinery, "C" is $29000.
Step: 2
The decay rate, "r" is 4% = 0.04
Step: 3
The exponential decay can be modeled by the equation y = C(1 - r)t.
Step: 4
y = 29000(1 - 0.04)t
  [Substitute 29000 for C, and 0.04 for r.]
Step: 5
y = 29000(0.96)t
  [Subtract 0.04 from 1.]
Step: 6
The exponential decay model that represents the depreciation of the machinery is y = 29000(0.96)t
Correct Answer is :   y = 29000(0.96)t

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