Jim invested \( \$ 5600 \) in an account that pays an annual interest rate of \( 2.2 \% \), compounded monthly. Answer e list of financial formulas. (a) Find the amount in the account after one year, assuming no withdrawals are made. Do not round any intermediate computations, and round your answer to the nearest cent. \( \$ \square \) (b) Find the effective annual interest rate, expressed as a percentage. Do not round any intermediate computations, and round your answer to the nearest hundredth of a percent. \( \square \% \)

To find the amount in the account after one year, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( A \) is the amount in the account after \( t \) years, - \( P \) is the principal amount (initial investment), - \( r \) is the annual interest rate, - \( n \) is the number of times the interest is compounded per year, - \( t \) is the time in years. Given: - \( P = \$5600 \), - \( r = 2.2\% = 0.022 \), - \( n = 12 \) (compounded monthly), - \( t = 1 \) year. Substitute the values into the formula: \[ A = 5600 \left(1 + \frac{0.022}{12}\right)^{12 \cdot 1} \] Now, we can calculate the amount in the account after one year. Calculate the value by following steps: - step0: Calculate: \(5600\left(1+\frac{0.022}{12}\right)^{12}\) - step1: Divide the terms: \(5600\left(1+\frac{11}{6000}\right)^{12}\) - step2: Add the numbers: \(5600\left(\frac{6011}{6000}\right)^{12}\) - step3: Simplify: \(5600\times \frac{6011^{12}}{6000^{12}}\) - step4: Rewrite the expression: \(400\times 14\times \frac{6011^{12}}{6000^{12}}\) - step5: Rewrite the expression: \(400\times 14\times \frac{6011^{12}}{400^{12}\times 15^{12}}\) - step6: Reduce the numbers: \(14\times \frac{6011^{12}}{400^{11}\times 15^{12}}\) - step7: Rewrite the expression: \(2\times 7\times \frac{6011^{12}}{400^{11}\times 15^{12}}\) - step8: Rewrite the expression: \(2\times 7\times \frac{6011^{12}}{16^{11}\times 25^{11}\times 15^{12}}\) - step9: Rewrite the expression: \(2\times 7\times \frac{6011^{12}}{2^{44}\times 25^{11}\times 15^{12}}\) - step10: Reduce the numbers: \(7\times \frac{6011^{12}}{2^{43}\times 25^{11}\times 15^{12}}\) - step11: Multiply the fractions: \(\frac{7\times 6011^{12}}{2^{43}\times 25^{11}\times 15^{12}}\) The amount in the account after one year is approximately $5724.45. Now, to find the effective annual interest rate, we can use the formula: \[ \text{Effective Annual Interest Rate} = \left(1 + \frac{r}{n}\right)^n - 1 \] Substitute the values into the formula: \[ \text{Effective Annual Interest Rate} = \left(1 + \frac{0.022}{12}\right)^{12} - 1 \] Now, we can calculate the effective annual interest rate. Calculate the value by following steps: - step0: Calculate: \(\left(1+\frac{0.022}{12}\right)^{12}-1\) - step1: Divide the terms: \(\left(1+\frac{11}{6000}\right)^{12}-1\) - step2: Add the numbers: \(\left(\frac{6011}{6000}\right)^{12}-1\) - step3: Rewrite the expression: \(\frac{6011^{12}}{6000^{12}}-1\) - step4: Reduce fractions to a common denominator: \(\frac{6011^{12}}{6000^{12}}-\frac{6000^{12}}{6000^{12}}\) - step5: Transform the expression: \(\frac{6011^{12}-6000^{12}}{6000^{12}}\) The effective annual interest rate is approximately 2.22%. Therefore, the answers are: (a) The amount in the account after one year is $5724.45. (b) The effective annual interest rate is 2.22%.