A new car is purchased for 16500 dollars. The value of the car depreciates at \( 5.75 \% \) per year. What will the value of the car be, to the nearest cent, after 5 years?

To find the value of the car after 5 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, \( P = 16500 \) dollars, - Annual depreciation rate, \( r = 5.75\% \), - Number of years, \( n = 5 \). Substitute the given values into the formula: \[ V = 16500 \times (1 - 0.0575)^5 \] Now, we can calculate the value of the car after 5 years. Calculate the value by following steps: - step0: Calculate: \(16500\left(1-0.0575\right)^{5}\) - step1: Subtract the numbers: \(16500\times 0.9425^{5}\) - step2: Convert the expressions: \(16500\left(\frac{377}{400}\right)^{5}\) - step3: Simplify: \(16500\times \frac{377^{5}}{400^{5}}\) - step4: Rewrite the expression: \(100\times 165\times \frac{377^{5}}{400^{5}}\) - step5: Rewrite the expression: \(100\times 165\times \frac{377^{5}}{100^{5}\times 4^{5}}\) - step6: Reduce the numbers: \(165\times \frac{377^{5}}{100^{4}\times 4^{5}}\) - step7: Rewrite the expression: \(5\times 33\times \frac{377^{5}}{100^{4}\times 4^{5}}\) - step8: Rewrite the expression: \(5\times 33\times \frac{377^{5}}{25^{4}\times 4^{4}\times 4^{5}}\) - step9: Rewrite the expression: \(5\times 33\times \frac{377^{5}}{5^{8}\times 4^{4}\times 4^{5}}\) - step10: Reduce the numbers: \(33\times \frac{377^{5}}{5^{7}\times 4^{4}\times 4^{5}}\) - step11: Multiply the fractions: \(\frac{33\times 377^{5}}{4^{9}\times 5^{7}}\) The value of the car after 5 years will be approximately $12271.30 to the nearest cent.