Question Meg invested \( \$ 16,000 \) in a savings account. If the interest rate is \( 6 \% \) per year, how much will be in the account in 5 years with monthly compounding? Round your answer to the nearest cent. Do NOT round until you calculate the final answer

To calculate the amount in the savings account after 5 years with monthly compounding, 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 (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the number of years the money is invested or borrowed. Given: - \( P = 16000 \) - \( r = 0.06 \) (6% as a decimal) - \( n = 12 \) (monthly compounding) - \( t = 5 \) Now, we can substitute these values into the formula: \[ A = 16000 \left(1 + \frac{0.06}{12}\right)^{12 \times 5} \] Let's calculate this step by step. 1. Calculate \( \frac{r}{n} \): \[ \frac{0.06}{12} = 0.005 \] 2. Calculate \( nt \): \[ 12 \times 5 = 60 \] 3. Calculate \( A \): \[ A = 16000 \left(1 + 0.005\right)^{60} \] 4. Calculate \( 1 + 0.005 \): \[ 1 + 0.005 = 1.005 \] 5. Now calculate \( (1.005)^{60} \): \[ (1.005)^{60} \] 6. Finally, multiply by \( 16000 \). Let's perform the calculations. Calculate the value by following steps: - step0: Calculate: \(16000\left(1+\frac{0.06}{12}\right)^{12\times 5}\) - step1: Divide the terms: \(16000\left(1+\frac{1}{200}\right)^{12\times 5}\) - step2: Add the numbers: \(16000\left(\frac{201}{200}\right)^{12\times 5}\) - step3: Multiply the numbers: \(16000\left(\frac{201}{200}\right)^{60}\) - step4: Simplify: \(16000\times \frac{201^{60}}{200^{60}}\) - step5: Rewrite the expression: \(200\times 80\times \frac{201^{60}}{200^{60}}\) - step6: Reduce the numbers: \(80\times \frac{201^{60}}{200^{59}}\) - step7: Rewrite the expression: \(40\times 2\times \frac{201^{60}}{200^{59}}\) - step8: Rewrite the expression: \(40\times 2\times \frac{201^{60}}{40^{59}\times 5^{59}}\) - step9: Reduce the numbers: \(2\times \frac{201^{60}}{40^{58}\times 5^{59}}\) - step10: Rewrite the expression: \(2\times \frac{201^{60}}{8^{58}\times 5^{58}\times 5^{59}}\) - step11: Rewrite the expression: \(2\times \frac{201^{60}}{2^{174}\times 5^{58}\times 5^{59}}\) - step12: Reduce the numbers: \(1\times \frac{201^{60}}{2^{173}\times 5^{58}\times 5^{59}}\) - step13: Multiply the fractions: \(\frac{201^{60}}{2^{173}\times 5^{117}}\) After performing the calculations, we find that the amount in the account after 5 years with monthly compounding is approximately: \[ A \approx 21581.602441 \] Rounding this to the nearest cent, the final amount in the account will be: \[ A \approx 21581.60 \] Thus, the total amount in the account after 5 years will be **\$21,581.60**.