Suppose that \( \$ 2000 \) is loaned at a rate of \( 11 \% \), compounded quarterly. Assuming that no payments are made, find the amount owed after 10 years. Do not round any intermediate computations, and round your answer to the nearest cent.

To find the amount owed after 10 years, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( A \) is the amount owed after \( t \) years, - \( P \) is the principal amount (initial loan 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: - \( P = \$2000 \), - \( r = 11\% = 0.11 \), - \( n = 4 \) (compounded quarterly), - \( t = 10 \) years. Substitute the given values into the formula: \[ A = 2000 \left(1 + \frac{0.11}{4}\right)^{4 \cdot 10} \] Now, we can calculate the amount owed after 10 years. Calculate the value by following steps: - step0: Calculate: \(2000\left(1+\frac{0.11}{4}\right)^{4\times 10}\) - step1: Divide the terms: \(2000\left(1+\frac{11}{400}\right)^{4\times 10}\) - step2: Add the numbers: \(2000\left(\frac{411}{400}\right)^{4\times 10}\) - step3: Multiply the numbers: \(2000\left(\frac{411}{400}\right)^{40}\) - step4: Simplify: \(2000\times \frac{411^{40}}{400^{40}}\) - step5: Rewrite the expression: \(400\times 5\times \frac{411^{40}}{400^{40}}\) - step6: Reduce the numbers: \(5\times \frac{411^{40}}{400^{39}}\) - step7: Rewrite the expression: \(5\times \frac{411^{40}}{25^{39}\times 16^{39}}\) - step8: Rewrite the expression: \(5\times \frac{411^{40}}{5^{78}\times 16^{39}}\) - step9: Reduce the numbers: \(1\times \frac{411^{40}}{5^{77}\times 16^{39}}\) - step10: Multiply the fractions: \(\frac{411^{40}}{5^{77}\times 16^{39}}\) The amount owed after 10 years is approximately $5919.75. Therefore, after 10 years, the amount owed on the loan of $2000 at an interest rate of 11% compounded quarterly, with no payments made, is approximately $5919.75.