A couple deposits \( \$ 24,000 \) into an account earning \( 4 \% \) annual interest for 15 years. Calculate the future value of the investment if the interest is compounded semiannually. Round your answer to the nearest cent. Formu

To calculate the future value of an investment with compound interest, we can use the formula: \[ FV = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( FV \) = future value of the investment - \( P \) = principal amount (initial deposit) - \( r \) = annual interest rate (in decimal) - \( n \) = number of times interest is compounded per year - \( t \) = number of years the money is invested or borrowed Given the values: - \( P = 24000 \) - \( r = 0.04 \) (4% as a decimal) - \( n = 2 \) (since the interest is compounded semiannually) - \( t = 15 \) Now, we can substitute these values into the formula to calculate the future value. Let's perform the calculation step by step. 1. Calculate \( \frac{r}{n} \): \[ \frac{r}{n} = \frac{0.04}{2} = 0.02 \] 2. Calculate \( nt \): \[ nt = 2 \times 15 = 30 \] 3. Substitute these values into the formula: \[ FV = 24000 \left(1 + 0.02\right)^{30} \] 4. Calculate \( 1 + 0.02 \): \[ 1 + 0.02 = 1.02 \] 5. Now calculate \( (1.02)^{30} \): \[ (1.02)^{30} \] 6. Finally, multiply by \( 24000 \) to find \( FV \). Let's perform the calculations. Calculate the value by following steps: - step0: Calculate: \(1.02^{30}\) - step1: Convert the expressions: \(\left(\frac{51}{50}\right)^{30}\) - step2: Use the properties of exponents: \(\frac{51^{30}}{50^{30}}\) Calculate or simplify the expression \( 24000 * (1.02^30) \). Calculate the value by following steps: - step0: Calculate: \(24000\times 1.02^{30}\) - step1: Convert the expressions: \(24000\left(\frac{51}{50}\right)^{30}\) - step2: Simplify: \(24000\times \frac{51^{30}}{50^{30}}\) - step3: Rewrite the expression: \(50\times 480\times \frac{51^{30}}{50^{30}}\) - step4: Reduce the numbers: \(480\times \frac{51^{30}}{50^{29}}\) - step5: Rewrite the expression: \(10\times 48\times \frac{51^{30}}{50^{29}}\) - step6: Rewrite the expression: \(10\times 48\times \frac{51^{30}}{10^{29}\times 5^{29}}\) - step7: Reduce the numbers: \(48\times \frac{51^{30}}{10^{28}\times 5^{29}}\) - step8: Rewrite the expression: \(16\times 3\times \frac{51^{30}}{10^{28}\times 5^{29}}\) - step9: Rewrite the expression: \(16\times 3\times \frac{51^{30}}{2^{28}\times 5^{28}\times 5^{29}}\) - step10: Rewrite the expression: \(2^{4}\times 3\times \frac{51^{30}}{2^{28}\times 5^{28}\times 5^{29}}\) - step11: Reduce the numbers: \(3\times \frac{51^{30}}{2^{24}\times 5^{28}\times 5^{29}}\) - step12: Multiply the fractions: \(\frac{3\times 51^{30}}{2^{24}\times 5^{57}}\) The calculations yield the following results: 1. \( (1.02)^{30} \approx 1.811362 \) 2. The future value \( FV \) is calculated as: \[ FV \approx 24000 \times 1.811362 \approx 43472.678018 \] Now, rounding this to the nearest cent, we find: \[ FV \approx 43472.68 \] Thus, the future value of the investment after 15 years, with interest compounded semiannually, is approximately **\$43,472.68**.