If 6600 dollars is invested at an interest rate of 5 percent per year, find the value of the investment at the end of 5 years for the following compounding methods. Round value to the nearest cent. (a) Annual: \( \$ \) (b) Semiannual: \( \$ \) (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 (initial investment). - \( r \) 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: - Initial investment \( P = \$6600 \) - Annual interest rate \( r = 5\% = 0.05 \) - Time \( t = 5 \) years We will calculate the value of the investment at the end of 5 years for each compounding method: (a) Annual compounding: \[ A = 6600 \left(1 + \frac{0.05}{1}\right)^{1 \times 5} \] (b) Semiannual compounding: \[ A = 6600 \left(1 + \frac{0.05}{2}\right)^{2 \times 5} \] (c) Monthly compounding: \[ A = 6600 \left(1 + \frac{0.05}{12}\right)^{12 \times 5} \] (d) Daily compounding: \[ A = 6600 \left(1 + \frac{0.05}{365}\right)^{365 \times 5} \] Let's calculate the values for each compounding method. Calculate the value by following steps: - step0: Calculate: \(6600\left(1+\frac{0.05}{1}\right)^{1}\times 5\) - step1: Divide the terms: \(6600\left(1+\frac{1}{20}\right)^{1}\times 5\) - step2: Add the numbers: \(6600\left(\frac{21}{20}\right)^{1}\times 5\) - step3: Calculate: \(6600\times \frac{21}{20}\times 5\) - step4: Multiply the terms: \(6930\times 5\) - step5: Multiply the numbers: \(34650\) Calculate or simplify the expression \( 6600*(1+0.05/2)^2*5 \). Calculate the value by following steps: - step0: Calculate: \(6600\left(1+\frac{0.05}{2}\right)^{2}\times 5\) - step1: Divide the terms: \(6600\left(1+\frac{1}{40}\right)^{2}\times 5\) - step2: Add the numbers: \(6600\left(\frac{41}{40}\right)^{2}\times 5\) - step3: Multiply the terms: \(33000\left(\frac{41}{40}\right)^{2}\) - step4: Evaluate the power: \(33000\times \frac{1681}{1600}\) - step5: Multiply: \(\frac{277365}{8}\) Calculate or simplify the expression \( 6600*(1+0.05/12)^12*5 \). Calculate the value by following steps: - step0: Calculate: \(6600\left(1+\frac{0.05}{12}\right)^{12}\times 5\) - step1: Divide the terms: \(6600\left(1+\frac{1}{240}\right)^{12}\times 5\) - step2: Add the numbers: \(6600\left(\frac{241}{240}\right)^{12}\times 5\) - step3: Multiply the terms: \(33000\left(\frac{241}{240}\right)^{12}\) - step4: Simplify: \(33000\times \frac{241^{12}}{240^{12}}\) - step5: Rewrite the expression: \(120\times 275\times \frac{241^{12}}{240^{12}}\) - step6: Rewrite the expression: \(120\times 275\times \frac{241^{12}}{120^{12}\times 2^{12}}\) - step7: Reduce the numbers: \(275\times \frac{241^{12}}{120^{11}\times 2^{12}}\) - step8: Rewrite the expression: \(25\times 11\times \frac{241^{12}}{120^{11}\times 2^{12}}\) - step9: Rewrite the expression: \(25\times 11\times \frac{241^{12}}{5^{11}\times 24^{11}\times 2^{12}}\) - step10: Rewrite the expression: \(5^{2}\times 11\times \frac{241^{12}}{5^{11}\times 24^{11}\times 2^{12}}\) - step11: Reduce the numbers: \(11\times \frac{241^{12}}{5^{9}\times 24^{11}\times 2^{12}}\) - step12: Multiply the fractions: \(\frac{11\times 241^{12}}{4096\times 5^{9}\times 24^{11}}\) Calculate or simplify the expression \( 6600*(1+0.05/365)^365*5 \). Calculate the value by following steps: - step0: Calculate: \(6600\left(1+\frac{0.05}{365}\right)^{365}\times 5\) - step1: Divide the terms: \(6600\left(1+\frac{1}{7300}\right)^{365}\times 5\) - step2: Add the numbers: \(6600\left(\frac{7301}{7300}\right)^{365}\times 5\) - step3: Multiply the terms: \(33000\left(\frac{7301}{7300}\right)^{365}\) - step4: Simplify: \(33000\times \frac{7301^{365}}{7300^{365}}\) - step5: Rewrite the expression: \(1000\times 33\times \frac{7301^{365}}{7300^{365}}\) - step6: Rewrite the expression: \(1000\times 33\times \frac{7301^{365}}{100^{365}\times 73^{365}}\) - step7: Rewrite the expression: \(10^{3}\times 33\times \frac{7301^{365}}{10^{730}\times 73^{365}}\) - step8: Reduce the numbers: \(33\times \frac{7301^{365}}{10^{727}\times 73^{365}}\) - step9: Multiply the fractions: \(\frac{33\times 7301^{365}}{10^{727}\times 73^{365}}\) The values of the investment at the end of 5 years for different compounding methods are: (a) Annual compounding: $34,650 (b) Semiannual compounding: $34,670.63 (c) Monthly compounding: $34,688.34 (d) Daily compounding: $34,691.83 Rounded to the nearest cent, the values are: (a) Annual compounding: $34,650 (b) Semiannual compounding: $34,670.63 (c) Monthly compounding: $34,688.34 (d) Daily compounding: $34,691.83