Question

a new car is purchased for 23200 dollars. the value of the car depreciates at 13.25% per year. what will the value of the car be, to the nearest cent, after 7 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 depreciation rate to decimal

The depreciation rate is $13.25\%$. To convert to decimal, divide by 100: $r = \frac{13.25}{100} = 0.1325$.

Step3: Substitute the values into the formula

We have $P = 23200$, $r = 0.1325$, and $t = 7$. Substitute into $V = P(1 - r)^t$:

$$
LATEXBLOCK0
$$

Step4: Calculate $(0.8675)^7$

First, calculate $0.8675^7$. Using a calculator, $0.8675^7 \approx 0.3642$.

Step5: Calculate the final value

Multiply by the initial price: $V = 23200\times0.3642 \approx 8450.04$.

Answer:

The value of the car after 7 years will be approximately $\$8450.04$.