A bank features a savings account that has an annual percentage rate of \( 3 \% \) with interest compounded monthly. Marissa deposits \( \$ 9,000 \) into the account. How much money will Marissa have in the account in 1 year? Answer \( =\$ \square \). Round answer to the nearest penny. What is the annual percentage yield (Apy for the savings account? APY \( =3.04 \) Question Help: Giin Calculator

To calculate the amount of money Marissa will have in the account in 1 year, 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 deposit). - \( r \) 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: - \( P = \$9,000 \) - \( r = 3\% = 0.03 \) - \( n = 12 \) (compounded monthly) - \( t = 1 \) year Substitute the values into the formula: \[ A = 9000 \left( 1 + \frac{0.03}{12} \right)^{12 \cdot 1} \] Now, we can calculate the amount of money Marissa will have in the account in 1 year. Calculate the value by following steps: - step0: Calculate: \(9000\left(1+\frac{0.03}{12}\right)^{12}\) - step1: Divide the terms: \(9000\left(1+\frac{1}{400}\right)^{12}\) - step2: Add the numbers: \(9000\left(\frac{401}{400}\right)^{12}\) - step3: Simplify: \(9000\times \frac{401^{12}}{400^{12}}\) - step4: Rewrite the expression: \(200\times 45\times \frac{401^{12}}{400^{12}}\) - step5: Rewrite the expression: \(200\times 45\times \frac{401^{12}}{200^{12}\times 2^{12}}\) - step6: Reduce the numbers: \(45\times \frac{401^{12}}{200^{11}\times 2^{12}}\) - step7: Rewrite the expression: \(5\times 9\times \frac{401^{12}}{200^{11}\times 2^{12}}\) - step8: Rewrite the expression: \(5\times 9\times \frac{401^{12}}{25^{11}\times 8^{11}\times 2^{12}}\) - step9: Rewrite the expression: \(5\times 9\times \frac{401^{12}}{5^{22}\times 8^{11}\times 2^{12}}\) - step10: Reduce the numbers: \(9\times \frac{401^{12}}{5^{21}\times 8^{11}\times 2^{12}}\) - step11: Multiply the fractions: \(\frac{9\times 401^{12}}{5^{21}\times 8^{15}}\) The amount of money Marissa will have in the account in 1 year is approximately $9,273.74. To find the annual percentage yield (APY), we can use the formula: \[ APY = \left( 1 + \frac{r}{n} \right)^n - 1 \] Substitute the values into the formula: \[ APY = \left( 1 + \frac{0.03}{12} \right)^{12} - 1 \] Now, we can calculate the APY. Calculate the value by following steps: - step0: Calculate: \(\left(1+\frac{0.03}{12}\right)^{12}-1\) - step1: Divide the terms: \(\left(1+\frac{1}{400}\right)^{12}-1\) - step2: Add the numbers: \(\left(\frac{401}{400}\right)^{12}-1\) - step3: Rewrite the expression: \(\frac{401^{12}}{400^{12}}-1\) - step4: Reduce fractions to a common denominator: \(\frac{401^{12}}{400^{12}}-\frac{400^{12}}{400^{12}}\) - step5: Transform the expression: \(\frac{401^{12}-400^{12}}{400^{12}}\) The annual percentage yield (APY) for the savings account is approximately 3.04%. Therefore, Marissa will have $9,273.74 in the account in 1 year, and the APY for the savings account is 3.04%.