A new car is purchased for 20300 dollars. The value of the car depreciates at \( 8.75 \% \) per year. What will the value of the car be, to the nearest cent, after 12 years?

To find the value of the car after 12 years, we can use the formula for depreciation: \[ V = P \times (1 - r)^n \] where: - \( V \) is the value of the car after \( n \) years, - \( P \) is the initial value of the car, - \( r \) is the annual depreciation rate, - \( n \) is the number of years. Given: - Initial value of the car, \( P = 20300 \) dollars, - Annual depreciation rate, \( r = 8.75\% \), - Number of years, \( n = 12 \). Substitute the given values into the formula: \[ V = 20300 \times (1 - 0.0875)^{12} \] Now, we can calculate the value of the car after 12 years. Calculate the value by following steps: - step0: Calculate: \(20300\left(1-0.0875\right)^{12}\) - step1: Subtract the numbers: \(20300\times 0.9125^{12}\) - step2: Convert the expressions: \(20300\left(\frac{73}{80}\right)^{12}\) - step3: Simplify: \(20300\times \frac{73^{12}}{80^{12}}\) - step4: Rewrite the expression: \(20\times 1015\times \frac{73^{12}}{80^{12}}\) - step5: Rewrite the expression: \(20\times 1015\times \frac{73^{12}}{20^{12}\times 4^{12}}\) - step6: Reduce the numbers: \(1015\times \frac{73^{12}}{20^{11}\times 4^{12}}\) - step7: Rewrite the expression: \(5\times 203\times \frac{73^{12}}{20^{11}\times 4^{12}}\) - step8: Rewrite the expression: \(5\times 203\times \frac{73^{12}}{5^{11}\times 4^{11}\times 4^{12}}\) - step9: Reduce the numbers: \(203\times \frac{73^{12}}{5^{10}\times 4^{11}\times 4^{12}}\) - step10: Multiply the fractions: \(\frac{203\times 73^{12}}{5^{10}\times 4^{23}}\) The value of the car after 12 years will be approximately $6765.35 to the nearest cent.