A bank features a savings account that has an annual percentage rate of \( 4 \% \) with interest compounded quarterly. Enzo deposits \( \$ 2,000 \) into the account. How much money will Enzo have in the account in 1 year? Answer \( =\$ \) Round answer to the nearest penny. What is the annual percentage yield (APY) for the savings account? APY \( =\square \) Round to the nearest hundredth of a percent.

To solve this problem, we will use the formula for compound interest and the formula for the annual percentage yield (APY). ### Step 1: Calculate the amount in the account after 1 year The formula for compound interest is given by: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) = the amount of money accumulated after n years, including interest. - \( P \) = the principal amount (the initial amount of money). - \( r \) = the annual interest rate (decimal). - \( n \) = the number of times that interest is compounded per year. - \( t \) = the number of years the money is invested or borrowed. Given: - \( P = 2000 \) - \( r = 0.04 \) (4% as a decimal) - \( n = 4 \) (quarterly compounding) - \( t = 1 \) Now, we can substitute these values into the formula: \[ A = 2000 \left(1 + \frac{0.04}{4}\right)^{4 \times 1} \] Let's calculate this step by step. ### Step 2: Calculate the amount \( A \) First, we calculate \( \frac{0.04}{4} \): \[ \frac{0.04}{4} = 0.01 \] Now, substitute this back into the formula: \[ A = 2000 \left(1 + 0.01\right)^{4} \] This simplifies to: \[ A = 2000 \left(1.01\right)^{4} \] Next, we calculate \( (1.01)^{4} \): \[ (1.01)^{4} = 1.04060401 \] Now, multiply by 2000: \[ A = 2000 \times 1.04060401 \] Now, let's calculate this value. ### Step 3: Calculate the final amount Now, I will perform the calculation for \( A \). Calculate the value by following steps: - step0: Calculate: \(2000\times 1.04060401\) - step1: Multiply the numbers: \(2081.20802\) The calculated amount \( A \) is approximately \( 2081.21 \) when rounded to the nearest penny. ### Final Amount in the Account \[ A \approx \$ 2081.21 \] ### Step 4: Calculate the Annual Percentage Yield (APY) The formula for the APY is given by: \[ APY = \left(1 + \frac{r}{n}\right)^{n} - 1 \] Where: - \( r \) = the annual interest rate (decimal). - \( n \) = the number of times that interest is compounded per year. Using the same values: - \( r = 0.04 \) - \( n = 4 \) Substituting these values into the APY formula: \[ APY = \left(1 + \frac{0.04}{4}\right)^{4} - 1 \] This simplifies to: \[ APY = \left(1 + 0.01\right)^{4} - 1 \] Now, we already calculated \( (1.01)^{4} \) as \( 1.04060401 \). So we can substitute this value: \[ APY = 1.04060401 - 1 \] Now, let's calculate the APY. Calculate the value by following steps: - step0: Calculate: \(1.04060401-1\) - step1: Subtract the numbers: \(0.04060401\) The calculated APY is approximately \( 0.04060401 \). ### Step 5: Convert APY to Percentage To express the APY as a percentage, we multiply by 100: \[ APY \approx 0.04060401 \times 100 \approx 4.060401 \] Rounding this to the nearest hundredth of a percent gives: \[ APY \approx 4.06\% \] ### Final Answers - The amount in the account after 1 year: \[ A \approx \$ 2081.21 \] - The Annual Percentage Yield (APY): \[ APY \approx 4.06\% \]