You deposit \( \$ 2000 \) in an account earning \( 4 \% \) interest compounded quarterly. How much will you have in the account in 15 years? Round your answer to the nearest cent.

To calculate the amount of money in the account after 15 years with compound interest, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (initial deposit). - \( r \) is the annual interest rate (in decimal form). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the time the money is invested for in years. Given: - \( P = \$2000 \) - \( r = 4\% = 0.04 \) - \( n = 4 \) (compounded quarterly) - \( t = 15 \) years Substitute the given values into the formula: \[ A = 2000 \left(1 + \frac{0.04}{4}\right)^{4 \times 15} \] Now, we can calculate the amount of money in the account after 15 years. Calculate the value by following steps: - step0: Calculate: \(2000\left(1+\frac{0.04}{4}\right)^{4\times 15}\) - step1: Divide the terms: \(2000\left(1+\frac{1}{100}\right)^{4\times 15}\) - step2: Add the numbers: \(2000\left(\frac{101}{100}\right)^{4\times 15}\) - step3: Multiply the numbers: \(2000\left(\frac{101}{100}\right)^{60}\) - step4: Simplify: \(2000\times \frac{101^{60}}{100^{60}}\) - step5: Rewrite the expression: \(1000\times 2\times \frac{101^{60}}{100^{60}}\) - step6: Rewrite the expression: \(10^{3}\times 2\times \frac{101^{60}}{10^{120}}\) - step7: Reduce the numbers: \(2\times \frac{101^{60}}{10^{117}}\) - step8: Rewrite the expression: \(2\times \frac{101^{60}}{2^{117}\times 5^{117}}\) - step9: Reduce the numbers: \(1\times \frac{101^{60}}{2^{116}\times 5^{117}}\) - step10: Multiply the fractions: \(\frac{101^{60}}{2^{116}\times 5^{117}}\) The amount of money in the account after 15 years with compound interest is approximately $3633.39. Rounding to the nearest cent, the amount is $3633.39.