To calculate the approximate value of the vehicle after 13 years, we can use the formula for exponential decay:
\[ V = P(1 - r)^t \]
where:
- \( V \) is the value of the vehicle after \( t \) years,
- \( P \) is the initial purchase price of the vehicle,
- \( r \) is the depreciation rate,
- \( t \) is the time in years.
In this case:
- \( P = 22400 \)
- \( r = 0.09 \)
- \( t = 13 \)
Now we can substitute these values into the formula:
\[ V = 22400(1 - 0.09)^{13} \]
Calculating \( (1 - 0.09) \):
\[ 1 - 0.09 = 0.91 \]
Now raise \( 0.91 \) to the power of 13:
\[ 0.91^{13} \approx 0.338 \]
Now multiply this by the initial value \( P \):
\[ V \approx 22400 \times 0.338 \approx 7563.20 \]
Thus, the approximate value of the vehicle 13 years after the purchase is $7,563.20.