Use the compound interest formula to compute the total amount accumulated and the interest eamed. \( \$ 4500 \) for 3 years at \( 8 \% \) compounded quarterly The total amount accumulated after 3 years is \( \$ \square \). (Round to the nearest cent as needed.)

To calculate the total amount accumulated and the interest earned using the compound interest formula, we can use the formula: \[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \] where: - \( A \) is the total amount accumulated - \( P \) is the principal amount (initial amount) - \( r \) is the annual interest rate (in decimal form) - \( n \) is the number of times the interest is compounded per year - \( t \) is the time in years Given: - Principal amount (\( P \)) = $4500 - Annual interest rate (\( r \)) = 8% = 0.08 - Compounded quarterly, so \( n = 4 \) - Time (\( t \)) = 3 years Substitute the given values into the formula: \[ A = 4500 \left( 1 + \frac{0.08}{4} \right)^{4 \times 3} \] Now, we can calculate the total amount accumulated after 3 years. Calculate the value by following steps: - step0: Calculate: \(4500\left(1+\frac{0.08}{4}\right)^{4\times 3}\) - step1: Divide the terms: \(4500\left(1+\frac{1}{50}\right)^{4\times 3}\) - step2: Add the numbers: \(4500\left(\frac{51}{50}\right)^{4\times 3}\) - step3: Multiply the numbers: \(4500\left(\frac{51}{50}\right)^{12}\) - step4: Simplify: \(4500\times \frac{51^{12}}{50^{12}}\) - step5: Rewrite the expression: \(50\times 90\times \frac{51^{12}}{50^{12}}\) - step6: Reduce the numbers: \(90\times \frac{51^{12}}{50^{11}}\) - step7: Rewrite the expression: \(10\times 9\times \frac{51^{12}}{50^{11}}\) - step8: Rewrite the expression: \(10\times 9\times \frac{51^{12}}{10^{11}\times 5^{11}}\) - step9: Reduce the numbers: \(9\times \frac{51^{12}}{10^{10}\times 5^{11}}\) - step10: Multiply the fractions: \(\frac{9\times 51^{12}}{10^{10}\times 5^{11}}\) The total amount accumulated after 3 years is approximately $5707.09. To find the interest earned, we can subtract the principal amount from the total amount accumulated: Interest earned = Total amount accumulated - Principal amount Interest earned = $5707.09 - $4500 Interest earned = $1207.09 Therefore, the interest earned after 3 years is $1207.09.