Question

a new car is purchased for 17600 dollars. the value of the car depreciates at 11.75% 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 (as a decimal), and $t$ is the time in years.

Step2: Convert the depreciation rate to a decimal

The depreciation rate is 11.75%, so $r = \frac{11.75}{100} = 0.1175$.

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

Here, $P = 17600$ dollars, $r = 0.1175$, and $t = 11$ years.

Step4: Substitute the values into the formula

Substitute into $V = P(1 - r)^t$:

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Step5: Calculate \((0.8825)^{11}\)

Using a calculator, $(0.8825)^{11}\approx0.2744$ (rounded to four decimal places).

Step6: Calculate the final value \( V \)

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Answer:

The value of the car after 11 years will be approximately $\boldsymbol{4829.44}$ dollars.