Question

a new car is purchased for 200000 dollars. the value of the car depreciates at 12.5% per year. what will the value of the car be, to the nearest cent, after 11 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 $12.5\%$, so $r = \frac{12.5}{100} = 0.125$.

Step3: Identify the values of \(P\), \(r\), and \(t\)

We have $P = 200000$, $r = 0.125$, and $t = 11$.

Step4: Substitute the values into the formula

\[

$$\begin{align*} V&=200000\times(1 - 0.125)^{11}\\ &=200000\times(0.875)^{11} \end{align*}$$

\]
First, calculate $(0.875)^{11}$. Using a calculator, $(0.875)^{11}\approx0.201414$.
Then, multiply by 200000: $V = 200000\times0.201414 = 40282.8$ (rounded to the nearest cent).

Answer:

The value of the car after 11 years will be $\$40282.80$.