Question

Write an exponential function for the situation below and use your equation to answer the question.

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Patricia purchases a car for $18, 995. The car depreciates at a rate of 18% annually. How much is the car worth in six years?

Answer 1

To model the depreciation of Patricia's car, we can use the formula for exponential decay:

\[
V(t) = V_0 (1 - r)^t
\]

Where:
- \( V(t) \) is the value of the car after \( t \) years.
- \( V_0 \) is the initial value of the car.
- \( r \) is the depreciation rate (expressed as a decimal).
- \( t \) is the number of years.

Given:
- \( V_0 = 18,995 \)
- \( r = 0.18 \)
- We want to find \( V(6) \) for \( t = 6 \).

Plugging in the values:

\[
V(6) = 18,995 \times (1 - 0.18)^6
\]

Simplifying the equation:

\[
V(6) = 18,995 \times (0.82)^6
\]

Now, calculate \( (0.82)^6 \):

\[
(0.82)^6 \approx 0.384432
\]

Now multiply by the initial value:

\[
V(6) \approx 18,995 \times 0.384432 \approx 7,292.36
\]

Thus, the value of the car after 6 years is approximately **$7,292.36**.