Which expression could complete the equation used to solve the problem?
Today, a motorcycle is valued at $35.000. The value is expected to decrease at a rate of 7.5% each year. What is the value of the motorcycle expected to be 5 years from now?
y=35,000.=_
a. (7.5)⁵
b. (1.075)⁵
c. (0.925)⁵
d. (92.5)⁵
e. (0.075)⁵
The Correct Answer and Explanation is :
To determine the value of the motorcycle five years from now, given that its current value is $35,000 and it depreciates at a rate of 7.5% per year, we need to utilize the formula for exponential decay. The value of an asset after a certain period of depreciation can be calculated using the formula:
[
V = P(1 – r)^t
]
where:
- ( V ) is the future value of the asset,
- ( P ) is the present value (initial value),
- ( r ) is the rate of depreciation (expressed as a decimal), and
- ( t ) is the number of years.
In this problem:
- The present value ( P = 35,000 )
- The depreciation rate ( r = 0.075 ) (which is 7.5% expressed as a decimal)
- The time ( t = 5 ) years
The first step is to compute ( 1 – r ):
[
1 – r = 1 – 0.075 = 0.925
]
Now, we substitute these values into the formula:
[
V = 35,000 \times (0.925)^5
]
This shows that the expression ( (0.925)^5 ) is crucial in calculating the motorcycle’s value in 5 years. The expression describes the remaining value of the motorcycle after 5 years of depreciation at 7.5% per year.
Now, let’s analyze the options given:
- a. (7.5)⁵: This does not represent depreciation; it’s simply raising the percentage to the fifth power.
- b. (1.075)⁵: This represents appreciation (not depreciation) and is incorrect in this context.
- c. (0.925)⁵: This correctly represents the depreciation over 5 years.
- d. (92.5)⁵: This is incorrect as it does not relate to depreciation at all.
- e. (0.075)⁵: This is also incorrect; it doesn’t relate to the depreciation calculation.
Therefore, the correct answer is c. (0.925)⁵.
In summary, to find the value of the motorcycle after 5 years, we would multiply the initial value of $35,000 by ( (0.925)^5 ), reflecting the impact of annual depreciation on its overall value.