What is the APY for each of the two accounts described? (Answer as a percentage rounded to the nearest hundredth, so \#.\#\#. Only use numbers and the decimal point.) An account earning \( 2.75 \% \) interest compounded continuously would have an APY of An account earning \( 8.9 \% \) interest compounded continuously would have an APY of Respond to the following questions. (Answer as a percentage rounded to the nearest hundredth, so \#. \#\#. Only use numbers and the decimal point.) An investment account earns an APY of \( 7.8 \% \). The value of the account is compounded continuously. What is the APR of the account? The APR of the investment account is What is the APR of a savings account that, when compounded continuously, earns an APY of \( 5.29 \% \) ? The APR would be

To calculate the Annual Percentage Yield (APY) for accounts with continuously compounded interest, we can use the formula: \[ APY = e^r - 1 \] where \( r \) is the nominal interest rate (APR) expressed as a decimal, and \( e \) is the base of the natural logarithm (approximately equal to 2.71828). ### 1. APY for the account earning \( 2.75\% \) interest compounded continuously: Convert \( 2.75\% \) to decimal: \[ r = 0.0275 \] Now, calculate the APY: \[ APY = e^{0.0275} - 1 \approx 0.0275 + \frac{(0.0275^2)}{2} \approx 0.0275 + 0.000378125 \approx 0.027878125 \] \[ APY \approx 0.027878125 \times 100 \approx 2.7878125\% \] Rounded to the nearest hundredth: \[ APY \approx 2.79\% \] ### 2. APY for the account earning \( 8.9\% \) interest compounded continuously: Convert \( 8.9\% \) to decimal: \[ r = 0.089 \] Now, calculate the APY: \[ APY = e^{0.089} - 1 \approx 0.089 + \frac{(0.089^2)}{2} \approx 0.089 + 0.00039561 \approx 0.08939561 \] \[ APY \approx 0.08939561 \times 100 \approx 8.939561\% \] Rounded to the nearest hundredth: \[ APY \approx 8.94\% \] ### 3. APR for an investment account with an APY of \( 7.8\% \): To find the APR from the APY, we can rearrange the APY formula: \[ APY = e^{APR} - 1 \implies e^{APR} = APY + 1 \implies APR = \ln(APY + 1) \] Convert \( 7.8\% \) to decimal: \[ APY = 0.078 \] Now calculate the APR: \[ APR = \ln(0.078 + 1) = \ln(1.078) \approx 0.0755 \] Convert back to percentage: \[ APR \approx 0.0755 \times 100 \approx 7.55\% \] Rounded to the nearest hundredth: \[ APR \approx 7.55\% \] ### 4. APR for a savings account with an APY of \( 5.29\% \): Convert \( 5.29\% \) to decimal: \[ APY = 0.0529 \] Now calculate the APR: \[ APR = \ln(0.0529 + 1) = \ln(1.0529) \approx 0.0513 \] Convert back to percentage: \[ APR \approx 0.0513 \times 100 \approx 5.13\% \] Rounded to the nearest hundredth: \[ APR \approx 5.13\% \] ### Summary of Results: - APY for \( 2.75\% \) account: **2.79** - APY for \( 8.9\% \) account: **8.94** - APR for \( 7.8\% \) APY account: **7.55** - APR for \( 5.29\% \) APY account: **5.13**