A new car is purchased for 23700 dollars. The value of the car depreciates at \( 13.5 \% \) per year. What will the value of the car be, to the nearest cent, after 10 years?

To find the value of the car after 10 years, we can use the formula for exponential decay, which is given by: \[ V = P(1 - r)^t \] where: - \( V \) is the future value of the car, - \( P \) is the initial value of the car (23700 dollars), - \( r \) is the rate of depreciation (13.5% or 0.135), - \( t \) is the time in years (10 years). Now, let's substitute the known values into the formula: \[ V = 23700(1 - 0.135)^{10} \] First, we need to calculate \( 1 - 0.135 \): \[ 1 - 0.135 = 0.865 \] Now we can calculate \( 0.865^{10} \): \[ V = 23700 \times 0.865^{10} \] Next, we will calculate \( 0.865^{10} \) and then multiply it by 23700. Let's perform these calculations. Calculate the value by following steps: - step0: Calculate: \(0.865^{10}\) - step1: Convert the expressions: \(\left(\frac{173}{200}\right)^{10}\) - step2: Use the properties of exponents: \(\frac{173^{10}}{200^{10}}\) Calculate or simplify the expression \( 23700 * 0.23451 \). Calculate the value by following steps: - step0: Calculate: \(23700\times 0.23451\) - step1: Multiply the numbers: \(5557.887\) Now that we have calculated \( 0.865^{10} \approx 0.23451 \), we can find the future value of the car: \[ V = 23700 \times 0.23451 \approx 5557.887 \] Rounding this to the nearest cent, the value of the car after 10 years will be: \[ \text{Value of the car} \approx 5557.89 \text{ dollars} \] Thus, the value of the car after 10 years is approximately **$5557.89**.