If 9200 dollars is invested at an interest rate of 6 percent per year, find the value of the investment at the end of 5 years for the following compounding methods, to the nearest cent. (a) Annual: \( \$ \) (b) Semiannual: \( \$ \square \) (c) Monthly: \( \$ \) (d) Daily: \( \$ \)

To find the value of the investment at the end of 5 years for different compounding methods, 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: - \( P = 9200 \) - \( r = 0.06 \) - \( t = 5 \) We will calculate \( A \) for each compounding method: 1. **Annual Compounding** (\( n = 1 \)) 2. **Semiannual Compounding** (\( n = 2 \)) 3. **Monthly Compounding** (\( n = 12 \)) 4. **Daily Compounding** (\( n = 365 \)) Let's calculate each case step by step. ### (a) Annual Compounding \[ A = 9200 \left(1 + \frac{0.06}{1}\right)^{1 \cdot 5} \] ### (b) Semiannual Compounding \[ A = 9200 \left(1 + \frac{0.06}{2}\right)^{2 \cdot 5} \] ### (c) Monthly Compounding \[ A = 9200 \left(1 + \frac{0.06}{12}\right)^{12 \cdot 5} \] ### (d) Daily Compounding \[ A = 9200 \left(1 + \frac{0.06}{365}\right)^{365 \cdot 5} \] Now, I will perform the calculations for each case. Calculate the value by following steps: - step0: Calculate: \(9200\left(1+\frac{0.06}{1}\right)^{1\times 5}\) - step1: Divide the terms: \(9200\left(1+\frac{3}{50}\right)^{1\times 5}\) - step2: Add the numbers: \(9200\left(\frac{53}{50}\right)^{1\times 5}\) - step3: Calculate: \(9200\left(\frac{53}{50}\right)^{5}\) - step4: Simplify: \(9200\times \frac{53^{5}}{50^{5}}\) - step5: Rewrite the expression: \(50\times 184\times \frac{53^{5}}{50^{5}}\) - step6: Reduce the numbers: \(184\times \frac{53^{5}}{50^{4}}\) - step7: Rewrite the expression: \(8\times 23\times \frac{53^{5}}{50^{4}}\) - step8: Rewrite the expression: \(8\times 23\times \frac{53^{5}}{2^{4}\times 25^{4}}\) - step9: Rewrite the expression: \(2^{3}\times 23\times \frac{53^{5}}{2^{4}\times 25^{4}}\) - step10: Reduce the numbers: \(23\times \frac{53^{5}}{2\times 25^{4}}\) - step11: Multiply the fractions: \(\frac{23\times 53^{5}}{2\times 25^{4}}\) Calculate or simplify the expression \( 9200 * (1 + 0.06/12)^(12*5) \). Calculate the value by following steps: - step0: Calculate: \(9200\left(1+\frac{0.06}{12}\right)^{12\times 5}\) - step1: Divide the terms: \(9200\left(1+\frac{1}{200}\right)^{12\times 5}\) - step2: Add the numbers: \(9200\left(\frac{201}{200}\right)^{12\times 5}\) - step3: Multiply the numbers: \(9200\left(\frac{201}{200}\right)^{60}\) - step4: Simplify: \(9200\times \frac{201^{60}}{200^{60}}\) - step5: Rewrite the expression: \(200\times 46\times \frac{201^{60}}{200^{60}}\) - step6: Reduce the numbers: \(46\times \frac{201^{60}}{200^{59}}\) - step7: Rewrite the expression: \(2\times 23\times \frac{201^{60}}{200^{59}}\) - step8: Rewrite the expression: \(2\times 23\times \frac{201^{60}}{8^{59}\times 25^{59}}\) - step9: Rewrite the expression: \(2\times 23\times \frac{201^{60}}{2^{177}\times 25^{59}}\) - step10: Reduce the numbers: \(23\times \frac{201^{60}}{2^{176}\times 25^{59}}\) - step11: Multiply the fractions: \(\frac{23\times 201^{60}}{2^{176}\times 25^{59}}\) Calculate or simplify the expression \( 9200 * (1 + 0.06/2)^(2*5) \). Calculate the value by following steps: - step0: Calculate: \(9200\left(1+\frac{0.06}{2}\right)^{2\times 5}\) - step1: Divide the terms: \(9200\left(1+\frac{3}{100}\right)^{2\times 5}\) - step2: Add the numbers: \(9200\left(\frac{103}{100}\right)^{2\times 5}\) - step3: Multiply the numbers: \(9200\left(\frac{103}{100}\right)^{10}\) - step4: Simplify: \(9200\times \frac{103^{10}}{100^{10}}\) - step5: Rewrite the expression: \(100\times 92\times \frac{103^{10}}{100^{10}}\) - step6: Reduce the numbers: \(92\times \frac{103^{10}}{100^{9}}\) - step7: Rewrite the expression: \(4\times 23\times \frac{103^{10}}{100^{9}}\) - step8: Rewrite the expression: \(4\times 23\times \frac{103^{10}}{4^{9}\times 25^{9}}\) - step9: Reduce the numbers: \(23\times \frac{103^{10}}{4^{8}\times 25^{9}}\) - step10: Multiply the fractions: \(\frac{23\times 103^{10}}{4^{8}\times 25^{9}}\) - step11: Evaluate the power: \(\frac{23\times 103^{10}}{65536\times 25^{9}}\) Calculate or simplify the expression \( 9200 * (1 + 0.06/365)^(365*5) \). Calculate the value by following steps: - step0: Calculate: \(9200\left(1+\frac{0.06}{365}\right)^{365\times 5}\) - step1: Divide the terms: \(9200\left(1+\frac{3}{18250}\right)^{365\times 5}\) - step2: Add the numbers: \(9200\left(\frac{18253}{18250}\right)^{365\times 5}\) - step3: Multiply the numbers: \(9200\left(\frac{18253}{18250}\right)^{1825}\) - step4: Simplify: \(9200\times \frac{18253^{1825}}{18250^{1825}}\) - step5: Rewrite the expression: \(50\times 184\times \frac{18253^{1825}}{18250^{1825}}\) - step6: Rewrite the expression: \(50\times 184\times \frac{18253^{1825}}{50^{1825}\times 365^{1825}}\) - step7: Reduce the numbers: \(184\times \frac{18253^{1825}}{50^{1824}\times 365^{1825}}\) - step8: Rewrite the expression: \(8\times 23\times \frac{18253^{1825}}{50^{1824}\times 365^{1825}}\) - step9: Rewrite the expression: \(8\times 23\times \frac{18253^{1825}}{2^{1824}\times 25^{1824}\times 365^{1825}}\) - step10: Rewrite the expression: \(2^{3}\times 23\times \frac{18253^{1825}}{2^{1824}\times 25^{1824}\times 365^{1825}}\) - step11: Reduce the numbers: \(23\times \frac{18253^{1825}}{2^{1821}\times 25^{1824}\times 365^{1825}}\) - step12: Multiply the fractions: \(\frac{23\times 18253^{1825}}{2^{1821}\times 25^{1824}\times 365^{1825}}\) Now, let's summarize the results for each compounding method: ### (a) Annual Compounding \[ A \approx 12311.68 \] ### (b) Semiannual Compounding \[ A \approx 12364.03 \] ### (c) Monthly Compounding \[ A \approx 12409.42 \] ### (d) Daily Compounding \[ A \approx 12418.39 \] Thus, the values of the investment at the end of 5 years for each compounding method are: - (a) Annual: \( \$12311.68 \) - (b) Semiannual: \( \$12364.03 \) - (c) Monthly: \( \$12409.42 \) - (d) Daily: \( \$12418.39 \)