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 an exponential function. Let's denote the initial value of the car as \( V_0 = \$18,000 \) and the depreciation rate as \( r = 12\% = 0.12 \). The exponential function to find the value of the car after \( t \) years is given by: \[ V(t) = V_0 \times (1 - r)^t \] Substitute the given values into the function: \[ V(t) = 18000 \times (1 - 0.12)^t \] Now, we can calculate the value of the car after 10 years by substituting \( t = 10 \) into the function: \[ V(10) = 18000 \times (1 - 0.12)^{10} \] Let's calculate this value. Calculate the value by following steps: - step0: Calculate: \(18000\left(1-0.12\right)^{10}\) - step1: Subtract the numbers: \(18000\times 0.88^{10}\) - step2: Convert the expressions: \(18000\left(\frac{22}{25}\right)^{10}\) - step3: Simplify: \(18000\times \frac{22^{10}}{25^{10}}\) - step4: Rewrite the expression: \(125\times 144\times \frac{22^{10}}{25^{10}}\) - step5: Rewrite the expression: \(5^{3}\times 144\times \frac{22^{10}}{5^{20}}\) - step6: Reduce the numbers: \(144\times \frac{22^{10}}{5^{17}}\) - step7: Multiply: \(\frac{144\times 22^{10}}{5^{17}}\) The value of the car after 10 years is approximately $5013.02.