Question

A used car was purchased for $12,329 in 2018 each in the car‘s value decreases 8.5% write an exponential function equation describing the situation what will the car be worth in 2024 on your answer to the nearest cent

Answer 1

To model the depreciation of the car's value, we can use an exponential decay function. The general form of an exponential decay function is:

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

where:
- \( V(t) \) is the value of the car after \( t \) years,
- \( V_0 \) is the initial value of the car,
- \( r \) is the rate of depreciation (as a decimal), and
- \( t \) is the number of years since the initial purchase.

In this case:
- \( V_0 = 12,329 \)
- \( r = 0.085 \) (which is 8.5% expressed as a decimal)

The equation describing the situation becomes:

\[
V(t) = 12,329 \cdot (1 - 0.085)^t
\]

Simplifying it, we have:

\[
V(t) = 12,329 \cdot (0.915)^t
\]

Next, we need to find the value of the car in 2024. Since the car was purchased in 2018, \( t \) for 2024 is:

\[
t = 2024 - 2018 = 6
\]

Now, we can substitute \( t = 6 \) into the equation to find the car's value in 2024:

\[
V(6) = 12,329 \cdot (0.915)^6
\]

Calculating \( (0.915)^6 \):

\[
(0.915)^6 \approx 0.577708
\]

Now, substituting back into the value equation:

\[
V(6) \approx 12,329 \cdot 0.577708 \approx 7,118.85
\]

Hence, the estimated value of the car in 2024 is approximately:

\[
\boxed{7,118.85}
\]

This means the car will be worth approximately $7,118.85 in 2024, rounded to the nearest cent.