Question

a new car is purchased for 19900 dollars. the value of the car depreciates at 7.5% 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 (in decimal), and \( t \) is the time in years.

Step2: Convert the rate to decimal

The depreciation rate \( r = 7.5\% = 0.075 \).

Step3: Substitute the values into the formula

We have \( P = 19900 \), \( r = 0.075 \), and \( t = 8 \). Plugging these into the formula: \( V = 19900(1 - 0.075)^8 \).
First, calculate \( 1 - 0.075 = 0.925 \). Then, calculate \( 0.925^8 \). Let's compute that: \( 0.925^8 \approx 0.5366 \) (using a calculator for the exponentiation). Then, multiply by 19900: \( V = 19900 \times 0.5366 \approx 10678.34 \).

Answer:

The value of the car after 8 years will be approximately \(\$10678.34\).