Question

a new car is purchased for 24800 dollars. the value of the car depreciates at 12% per year. what will the value of the car be, to the nearest cent, after 13 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 rate to a decimal

The depreciation rate is 12%, so $r = \frac{12}{100} = 0.12$.

Step3: Substitute the values into the formula

We have $P = 24800$, $r = 0.12$, and $t = 13$. Substituting these into the formula gives:
\[
V = 24800(1 - 0.12)^{13}
\]
\[
V = 24800(0.88)^{13}
\]

Step4: Calculate $(0.88)^{13}$

First, calculate $(0.88)^{13}$. Using a calculator, $(0.88)^{13}\approx0.1896907$.

Step5: Calculate the final value

Multiply this by the initial price:
\[
V = 24800\times0.1896907
\]
\[
V\approx24800\times0.1896907 = 4704.32936
\]
Rounding to the nearest cent (two decimal places), we get $V\approx4704.33$.

Answer:

The value of the car after 13 years will be approximately $\$4704.33$.