Suppose an investment account is opened with an intial deposit of \( \$ 15,000 \) earning \( 9.2 \% \) interest. Round all answers to the nearest dollar. a. How much will the account be worth after 20 years if it is compounded monthly? \( \$ \) b. How much will the account be worth after 20 years if it is compounded continuously? \( \$ \)

To calculate the future value of the investment account after 20 years with different compounding methods, we can use the formula for compound interest: \[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \] where: - \( A \) is the future value of the investment - \( P \) is the principal amount (initial deposit) - \( r \) is the annual interest rate - \( n \) is the number of times the interest is compounded per year - \( t \) is the time in years Given: - Initial deposit \( P = \$15,000 \) - Annual interest rate \( r = 9.2\% = 0.092 \) - Compounding frequency: a. Monthly compounding: \( n = 12 \) b. Continuous compounding: \( n = \infty \) a. For monthly compounding: \[ A = 15000 \left( 1 + \frac{0.092}{12} \right)^{12 \times 20} \] b. For continuous compounding: \[ A = 15000 \left( 1 + 0.092 \right)^{20} \] Let's calculate the future value for both scenarios. Calculate the value by following steps: - step0: Calculate: \(15000\left(1+\frac{0.092}{12}\right)^{12}\times 20\) - step1: Divide the terms: \(15000\left(1+\frac{23}{3000}\right)^{12}\times 20\) - step2: Add the numbers: \(15000\left(\frac{3023}{3000}\right)^{12}\times 20\) - step3: Multiply the terms: \(300000\left(\frac{3023}{3000}\right)^{12}\) - step4: Simplify: \(300000\times \frac{3023^{12}}{3000^{12}}\) - step5: Rewrite the expression: \(3000\times 100\times \frac{3023^{12}}{3000^{12}}\) - step6: Reduce the numbers: \(100\times \frac{3023^{12}}{3000^{11}}\) - step7: Rewrite the expression: \(100\times \frac{3023^{12}}{1000^{11}\times 3^{11}}\) - step8: Rewrite the expression: \(10^{2}\times \frac{3023^{12}}{10^{33}\times 3^{11}}\) - step9: Reduce the numbers: \(1\times \frac{3023^{12}}{10^{31}\times 3^{11}}\) - step10: Multiply the fractions: \(\frac{3023^{12}}{10^{31}\times 3^{11}}\) Calculate or simplify the expression \( 15000 * (1 + 0.092)^20 \). Calculate the value by following steps: - step0: Calculate: \(15000\left(1+0.092\right)^{20}\) - step1: Add the numbers: \(15000\times 1.092^{20}\) - step2: Convert the expressions: \(15000\left(\frac{273}{250}\right)^{20}\) - step3: Simplify: \(15000\times \frac{273^{20}}{250^{20}}\) - step4: Rewrite the expression: \(250\times 60\times \frac{273^{20}}{250^{20}}\) - step5: Reduce the numbers: \(60\times \frac{273^{20}}{250^{19}}\) - step6: Rewrite the expression: \(10\times 6\times \frac{273^{20}}{250^{19}}\) - step7: Rewrite the expression: \(10\times 6\times \frac{273^{20}}{10^{19}\times 25^{19}}\) - step8: Reduce the numbers: \(6\times \frac{273^{20}}{10^{18}\times 25^{19}}\) - step9: Rewrite the expression: \(2\times 3\times \frac{273^{20}}{10^{18}\times 25^{19}}\) - step10: Rewrite the expression: \(2\times 3\times \frac{273^{20}}{2^{18}\times 5^{18}\times 25^{19}}\) - step11: Reduce the numbers: \(3\times \frac{273^{20}}{2^{17}\times 5^{18}\times 25^{19}}\) - step12: Multiply the fractions: \(\frac{3\times 273^{20}}{2^{17}\times 5^{56}}\) a. The account will be worth approximately $32,879.41 after 20 years if it is compounded monthly. b. The account will be worth approximately $87,205.53 after 20 years if it is compounded continuously.