Question 13, 8.2.57 HW Score: $2222 \%, 4$ of 18 points Points: 0 of 1 Save Ulison is 27 years old and plans to retire at age 65 with $\$ 1,080,000$ in her retirement account. What amount would she have to set aside now in an investment paying $5 \%$ annual terest if the compounding is done daily (assume 365 days in a year)? he amount to be invested now is $\$ \square$ Round to the nearest cent as needed)
Solution
Step 1 ：Given that Ulison is 27 years old and plans to retire at age 65 with $1,080,000 in her retirement account.
Step 2 ：She wants to know how much she would have to set aside now in an investment paying 5% annual interest if the compounding is done daily (assume 365 days in a year).
Step 3 ：We can use the formula for the present value (PV) of a future sum (FV) compounded daily: \(PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}\), where r is the annual interest rate, n is the number of compounding periods in a year, and t is the time in years.
Step 4 ：Substitute the given values into the formula: \(PV = \frac{1,080,000}{(1 + \frac{0.05}{365})^{365 \times (65 - 27)}}\).
Step 5 ：Solving the equation gives the present value as approximately $161,555.13.
Step 6 ：Final Answer: The amount to be invested now is \(\boxed{\$161,555.13}\).
