Arnett College predicts that in 18 years it will take \( \$ 350,000 \) to attend the college for four years. Hailey has a substantial amount of cash and wishes to invest a lump sum of money for her child's college fund. How much should Hailey put aside in an account with an APR of \( 11 \% \) compounded monthly in order to have \( \$ 350,000 \) in the account in 18 years? Round your answer to the nearest cent, if necessary.

To determine how much Hailey should invest today to have \( \$350,000 \) in 18 years with an annual percentage rate (APR) of \( 11\% \) compounded monthly, we can use the formula for the present value of a future sum: \[ PV = \frac{FV}{(1 + r/n)^{nt}} \] Where: - \( PV \) = present value (the amount to invest today) - \( FV \) = future value (the amount desired in the future, which is \( \$350,000 \)) - \( r \) = annual interest rate (as a decimal, so \( 0.11 \) for \( 11\% \)) - \( n \) = number of times the interest is compounded per year (monthly compounding means \( n = 12 \)) - \( t \) = number of years the money is invested (which is \( 18 \) years) Now, we can plug in the values: - \( FV = 350,000 \) - \( r = 0.11 \) - \( n = 12 \) - \( t = 18 \) First, we calculate \( r/n \): \[ r/n = \frac{0.11}{12} \approx 0.00916667 \] Next, we calculate \( nt \): \[ nt = 12 \times 18 = 216 \] Now we can substitute these values into the present value formula: \[ PV = \frac{350,000}{(1 + 0.00916667)^{216}} \] Calculating \( (1 + 0.00916667)^{216} \): \[ (1 + 0.00916667)^{216} \approx (1.00916667)^{216} \approx 6.022575 \] Now we can calculate \( PV \): \[ PV = \frac{350,000}{6.022575} \approx 58,157.63 \] Thus, Hailey should invest approximately \( \boxed{58157.63} \) today to have \( \$350,000 \) in 18 years.