Allison is 33 years old and plans to retire at age 65 with \( \$ 1,390,000 \) in her retirement account. What amount would she have to set aside now in an investment paying \( 5 \% \) annual interest if the compounding is done daily (assume 365 days in a year)? The amount to be invested now is \( \$ \square \). (Round to the nearest cent as needed.)

To find the amount Allison needs to set aside now in an investment paying 5% 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 $1,390,000 in her retirement account. - The annual interest rate is 5% (0.05 in decimal form). - The compounding is done daily, so \( n = 365 \) days in a year. - The time the money is invested for is 32 years (from age 33 to 65). We need to find the principal amount \( P \) that Allison needs to set aside now. Substitute the given values into the formula: \[ 1390000 = P \left( 1 + \frac{0.05}{365} \right)^{365 \times 32} \] Now, we can solve for \( P \) to find the amount Allison needs to set aside now. Solve the equation \( 1390000 = P * (1 + 0.05/365)^365*32 \). Solve the equation by following steps: - step0: Solve for \(P\): \(1390000=P\left(1+\frac{0.05}{365}\right)^{365}\times 32\) - step1: Simplify: \(1390000=\frac{7301^{365}}{2^{725}\times 1825^{365}}\times P\) - step2: Swap the sides: \(\frac{7301^{365}}{2^{725}\times 1825^{365}}\times P=1390000\) - step3: Rewrite the expression: \(\frac{7301^{365}P}{2^{725}\times 1825^{365}}=1390000\) - step4: Cross multiply: \(7301^{365}P=2^{725}\times 1825^{365}\times 1390000\) - step5: Simplify the equation: \(7301^{365}P=1390000\times 2^{725}\times 1825^{365}\) - step6: Divide both sides: \(\frac{7301^{365}P}{7301^{365}}=\frac{1390000\times 2^{725}\times 1825^{365}}{7301^{365}}\) - step7: Divide the numbers: \(P=\frac{1390000\times 2^{725}\times 1825^{365}}{7301^{365}}\) The amount Allison needs to set aside now is approximately $\\frac{1390000\\times 2^{725}\\times 1825^{365}}{7301^{365}}$.