Question
The value v(t) of a car depreciates according to the function v(t) = P(0.85) t , where P is the purchase price of the car and t is the time, in years, since the car was purchased. By what percentage does the value of the car depreciate each year?
Answers
Answered by
GPT-4o mini
The function given for the value of the car is \( v(t) = P(0.85)^t \), where \( P \) is the purchase price and \( t \) is the time in years.
In this equation, \( 0.85 \) represents the proportion of the car's value that remains after each year. To find the percentage depreciation, we can calculate how much value is lost each year.
Since the value of the car after one year, \( v(1) \), can be expressed as:
\[
v(1) = P(0.85)^1 = 0.85P
\]
This indicates that after one year, the value of the car is \( 85\% \) of its original value \( P \). Therefore, the depreciation for the first year can be calculated as follows:
\[
\text{Depreciation} = P - v(1) = P - 0.85P = 0.15P
\]
To express this depreciation as a percentage of the original purchase price \( P \):
\[
\text{Percentage depreciation} = \frac{\text{Depreciation}}{P} \times 100 = \frac{0.15P}{P} \times 100 = 15\%
\]
Thus, the value of the car depreciates by **15%** each year.
In this equation, \( 0.85 \) represents the proportion of the car's value that remains after each year. To find the percentage depreciation, we can calculate how much value is lost each year.
Since the value of the car after one year, \( v(1) \), can be expressed as:
\[
v(1) = P(0.85)^1 = 0.85P
\]
This indicates that after one year, the value of the car is \( 85\% \) of its original value \( P \). Therefore, the depreciation for the first year can be calculated as follows:
\[
\text{Depreciation} = P - v(1) = P - 0.85P = 0.15P
\]
To express this depreciation as a percentage of the original purchase price \( P \):
\[
\text{Percentage depreciation} = \frac{\text{Depreciation}}{P} \times 100 = \frac{0.15P}{P} \times 100 = 15\%
\]
Thus, the value of the car depreciates by **15%** each year.