Example 5: The value of a car is \( \$ 18,000 \) and depreciating at a rate of \( 12 \% \) per year. Use an exponential function to find the value of the car after 10 years.

To find the value of the car after 10 years, we can use the exponential decay formula: \[ V(t) = V_0 \cdot (1 - r)^t \] where \( V_0 \) is the initial value, \( r \) is the rate of depreciation, and \( t \) is the number of years. Here, \( V_0 = 18000 \), \( r = 0.12 \), and \( t = 10 \). Plugging in the values: \[ V(10) = 18000 \cdot (1 - 0.12)^{10} = 18000 \cdot (0.88)^{10} \] Calculating \( (0.88)^{10} \) gives approximately \( 0.314 \), so: \[ V(10) \approx 18000 \cdot 0.314 \approx 5652 \] Thus, the value of the car after 10 years is approximately \( \$5,652 \).