Question

a new car is purchased for 16600 dollars. the value of the car depreciates at 9.75% per year. what will the value of the car be, to the nearest cent, after 8 years?

Explanation:

Step1: Identify the formula for depreciation

The formula for exponential depreciation is \( V = P(1 - r)^t \), where \( V \) is the final value, \( P \) is the initial principal (purchase price), \( r \) is the annual depreciation rate (as a decimal), and \( t \) is the time in years.

Step2: Convert the depreciation rate to a decimal

The depreciation rate is \( 9.75\% \), so \( r = \frac{9.75}{100} = 0.0975 \).

Step3: Substitute the values into the formula

We have \( P = 16600 \), \( r = 0.0975 \), and \( t = 8 \). Substituting these into the formula gives:

$$ V = 16600(1 - 0.0975)^8 $$

Step4: Calculate \( (1 - 0.0975) \)

\( 1 - 0.0975 = 0.9025 \).

Step5: Calculate \( 0.9025^8 \)

Using a calculator, \( 0.9025^8 \approx 0.4709 \) (rounded to four decimal places).

Step6: Calculate the final value \( V \)

Multiply \( 16600 \) by \( 0.4709 \):

$$ V = 16600 \times 0.4709 \approx 7816.94 $$

Answer:

The value of the car after 8 years will be approximately \(\$7816.94\).