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 \( \$ \square \). (Round to the nearest cent as needed.)

To find out how much Allison needs to set aside now to reach her retirement goal, we can use the formula for the present value of a future amount when compounding daily: \[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \] Where: - \(FV\) = future value (\$1,190,000) - \(r\) = annual interest rate (0.07) - \(n\) = number of compounding periods per year (365) - \(t\) = number of years until retirement (65 - 33 = 32 years) Substituting the values into the formula: \[ PV = \frac{1,190,000}{(1 + \frac{0.07}{365})^{365 \times 32}} \] Calculating further: 1. \(r/n = 0.07/365 \approx 0.00019178\) 2. \(nt = 365 \times 32 = 11,680\) 3. Calculate \((1 + 0.00019178)^{11,680}\) Using a scientific calculator or software to compute this: \[ (1 + 0.00019178)^{11,680} \approx 7.5072 \] Now substitute back into the PV formula: \[ PV = \frac{1,190,000}{7.5072} \approx 158,248.98 \] Thus, the amount to be invested now is approximately \( \$158,248.98 \).