Joanne deposits \( \$ 4300 \) into a one-year CD at a rate of \( 4.3 \% \) interest compounded daily. 1) What is her ending balance after the year? \( \$ \) 2) How much interest does she earn? \( \$ \) 3) What is her APY to two decimals as a percent?

To find Joanne's ending balance after the year, we can use the formula for compound interest: \[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \] where: - \( A \) is the ending balance - \( P \) is the principal amount (initial deposit) - \( r \) is the annual interest rate - \( n \) is the number of times interest is compounded per year - \( t \) is the time in years Given: - \( P = \$4300 \) - \( r = 4.3\% = 0.043 \) - \( n = 365 \) (compounded daily) - \( t = 1 \) year Substitute the values into the formula to find the ending balance: \[ A = 4300 \left( 1 + \frac{0.043}{365} \right)^{365 \cdot 1} \] Now, let's calculate the ending balance. Calculate the value by following steps: - step0: Calculate: \(4300\left(1+\frac{0.043}{365}\right)^{365}\) - step1: Divide the terms: \(4300\left(1+\frac{43}{365000}\right)^{365}\) - step2: Add the terms: \(4300\left(\frac{365043}{365000}\right)^{365}\) - step3: Simplify: \(4300\times \frac{365043^{365}}{365000^{365}}\) - step4: Rewrite the expression: \(100\times 43\times \frac{365043^{365}}{365000^{365}}\) - step5: Rewrite the expression: \(100\times 43\times \frac{365043^{365}}{1000^{365}\times 365^{365}}\) - step6: Rewrite the expression: \(10^{2}\times 43\times \frac{365043^{365}}{10^{1095}\times 365^{365}}\) - step7: Reduce the numbers: \(43\times \frac{365043^{365}}{10^{1093}\times 365^{365}}\) - step8: Multiply the fractions: \(\frac{43\times 365043^{365}}{10^{1093}\times 365^{365}}\) Joanne's ending balance after the year is approximately $4488.92. To find the interest earned, we subtract the principal amount from the ending balance: Interest = Ending Balance - Principal Amount Interest = $4488.92 - $4300 Interest = $188.92 Therefore, Joanne earns $188.92 in interest. To find the APY (Annual Percentage Yield) to two decimals as a percent, we can use the formula: \[ APY = \left( 1 + \frac{r}{n} \right)^{n} - 1 \] Substitute the values into the formula to find the APY: \[ APY = \left( 1 + \frac{0.043}{365} \right)^{365} - 1 \] Now, let's calculate the APY. Calculate the value by following steps: - step0: Calculate: \(\left(1+\frac{0.043}{365}\right)^{365}-1\) - step1: Divide the terms: \(\left(1+\frac{43}{365000}\right)^{365}-1\) - step2: Add the terms: \(\left(\frac{365043}{365000}\right)^{365}-1\) - step3: Rewrite the expression: \(\frac{365043^{365}}{365000^{365}}-1\) - step4: Reduce fractions to a common denominator: \(\frac{365043^{365}}{365000^{365}}-\frac{365000^{365}}{365000^{365}}\) - step5: Transform the expression: \(\frac{365043^{365}-365000^{365}}{365000^{365}}\) The APY to two decimals as a percent is approximately 4.39%. Therefore, Joanne's APY is 4.39%.