To determine the approximate value of the vehicle after 13 years given a purchase price of $22,400 and a depreciation rate of 9% per year, we can use the formula for exponential decay:
\[ V = P(1 - r)^t \]
where:
- \( V \) is the value after \( t \) years,
- \( P \) is the initial value ($22,400),
- \( r \) is the depreciation rate (9% or 0.09),
- \( t \) is the number of years (13).
Substituting the values into the formula:
\[ V = 22400(1 - 0.09)^{13} \]
First, calculate \( (1 - 0.09) \):
\[ 1 - 0.09 = 0.91 \]
Next, raise 0.91 to the power of 13:
\[ 0.91^{13} \approx 0.300129 \]
Now, multiply this value by the initial price:
\[ V \approx 22400 \times 0.300129 \approx 6729.4 \]
Rounding to the nearest whole number:
\[ V \approx 6730 \]
Thus, the approximate value of the vehicle after 13 years is $6,730.