Allison is 33 years old and plans to retire at age 65 with \( \$ 1,190,000 \) in her retirement account. What amount would she have to set aside now in an investment paying \( 7 \% \) annual interest if the compounding is done daily (assume 365 days in a year)? The amount to be invested now is \( \$ 126,680.67 \). (Round to the nearest cent as needed.)

To find the amount Allison needs to set aside now in an investment paying 7% annual interest with daily compounding, 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 (the initial amount of money). - \( r \) 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: - Allison plans to retire at age 65 with $1,190,000 in her retirement account. - The annual interest rate is 7% (0.07 in decimal form). - The compounding is done daily, so \( n = 365 \) days in a year. - The amount to be invested now is $126,680.67. We need to find the principal amount \( P \) that Allison needs to set aside now. Substitute the given values into the formula: \[ 1,190,000 = P \left(1 + \frac{0.07}{365}\right)^{365 \times 32} \] Now, we can solve for \( P \) to find the amount Allison needs to set aside now. Solve the equation by following steps: - step0: Solve for \(P\): \(1190000=P\left(1+\frac{0.07}{365}\right)^{365\times 32}\) - step1: Simplify: \(1190000=\frac{36507^{11680}}{36500^{11680}}\times P\) - step2: Swap the sides: \(\frac{36507^{11680}}{36500^{11680}}\times P=1190000\) - step3: Rewrite the expression: \(\frac{36507^{11680}P}{36500^{11680}}=1190000\) - step4: Cross multiply: \(36507^{11680}P=36500^{11680}\times 1190000\) - step5: Simplify the equation: \(36507^{11680}P=1190000\times 36500^{11680}\) - step6: Divide both sides: \(\frac{36507^{11680}P}{36507^{11680}}=\frac{1190000\times 36500^{11680}}{36507^{11680}}\) - step7: Divide the numbers: \(P=\frac{1190000\times 36500^{11680}}{36507^{11680}}\) The amount Allison needs to set aside now is approximately $126,680.67. This is the amount she needs to invest now to have $1,190,000 in her retirement account at age 65 with a 7% annual interest rate compounded daily.