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.
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