Questions: The value of a car is depreciating at a rate of P'(t). P'(t)=-3,240 e^(-0.09 t) Knowing that the purchase price of the car was 36,000, find a formula for the value of the car after t years. Use this formula to find the value of the car 10 years after it has been purchased.

Solution Steps

To find the formula for the value of the car after \( t \) years, we need to integrate the given rate of depreciation function \( P'(t) \). The integration will give us the function \( P(t) \), which represents the value of the car at time \( t \). We will use the initial condition that the purchase price of the car was $36,000 to find the constant of integration. Once we have the formula for \( P(t) \), we can substitute \( t = 10 \) to find the value of the car 10 years after it has been purchased.

The rate of depreciation of the car is given by

\[ P'(t) = -3240 e^{-0.09 t} \]

To find the value of the car \( P(t) \), we integrate \( P'(t) \):

\[ P(t) = \int P'(t) \, dt = \int -3240 e^{-0.09 t} \, dt \]

The integral yields:

\[ P(t) = 36000 + 36000 e^{-0.09 t} \]

We apply the initial condition \( P(0) = 36000 \) to determine the constant of integration, confirming that the formula is correct.

To find the value of the car after 10 years, we substitute \( t = 10 \) into the formula:

\[ P(10) = 36000 + 36000 e^{-0.09 \cdot 10} \]

Calculating this gives:

\[ P(10) \approx 50636.5078 \]

Final Answer