A new car is purchased for 15500 dollars. The value of the car depreciates at \( 7.5 \% \) per year. What will the value of the car be, to the nearest cent, after 6 years?

To find the value of the car after 6 years, we can use the formula for depreciation: \[ \text{Value after depreciation} = \text{Initial value} \times (1 - \text{Depreciation rate})^{\text{Number of years}} \] Given: - Initial value of the car = $15500 - Depreciation rate = 7.5% per year - Number of years = 6 years Substitute the values into the formula: \[ \text{Value after depreciation} = 15500 \times (1 - 0.075)^6 \] Now, we can calculate the value of the car after 6 years. Calculate the value by following steps: - step0: Calculate: \(15500\left(1-0.075\right)^{6}\) - step1: Subtract the numbers: \(15500\times 0.925^{6}\) - step2: Convert the expressions: \(15500\left(\frac{37}{40}\right)^{6}\) - step3: Simplify: \(15500\times \frac{37^{6}}{40^{6}}\) - step4: Rewrite the expression: \(20\times 775\times \frac{37^{6}}{40^{6}}\) - step5: Rewrite the expression: \(20\times 775\times \frac{37^{6}}{20^{6}\times 2^{6}}\) - step6: Reduce the numbers: \(775\times \frac{37^{6}}{20^{5}\times 2^{6}}\) - step7: Rewrite the expression: \(25\times 31\times \frac{37^{6}}{20^{5}\times 2^{6}}\) - step8: Rewrite the expression: \(25\times 31\times \frac{37^{6}}{5^{5}\times 4^{5}\times 2^{6}}\) - step9: Rewrite the expression: \(5^{2}\times 31\times \frac{37^{6}}{5^{5}\times 4^{5}\times 2^{6}}\) - step10: Reduce the numbers: \(31\times \frac{37^{6}}{5^{3}\times 4^{5}\times 2^{6}}\) - step11: Multiply: \(\frac{31\times 37^{6}}{8192000}\) The value of the car after 6 years will be approximately $9709.17 to the nearest cent.