A family is getting a divorce, and part of the divorce settlement involves setting aside money today for college tuition for their daughter who enters college in 9 years. They estimate that the cost of four years' tuition, food, and lodging at the state university their daughter will attend will be \( \$ 31,200 \). (a) Find the lump sum that must be invested at \( 4 \% \) semiannually. (b) Find the amount of interest earned. Click here to view periods \( 1-25 \) of the present value of a dollar table: Click here to view periods \( 26-49 \) of the present value of a dollar table. What is the present value? \( \$ \square \) (Round to the nearest cent as needed.) What is the amount of interest earned? \( \$ \square \) (Round to the nearest cent as needed.)

Given: - The cost of four years' tuition, food, and lodging at the state university is $31,200. - The family wants to set aside money today for college tuition for their daughter who enters college in 9 years. - The interest rate is 4% semiannually. (a) To find the lump sum that must be invested at 4% semiannually, we can use the present value formula: \[ PV = \frac{FV}{(1 + r)^n} \] where: - PV = present value (the amount of money to be invested today) - FV = future value (the cost of four years' tuition, food, and lodging) - r = interest rate per period - n = number of periods Given: - FV = $31,200 - r = 4% semiannually = 0.04/2 = 0.02 - n = 9 years * 2 (since the interest is compounded semiannually) = 18 periods Substitute the values into the formula: \[ PV = \frac{31,200}{(1 + 0.02)^{18}} \] Calculate the present value. (b) To find the amount of interest earned, we can subtract the present value from the future value: \[ Interest = FV - PV \] Calculate the interest earned. Calculate the value by following steps: - step0: Calculate: \(\frac{31200}{\left(1+0.02\right)^{18}}\) - step1: Add the numbers: \(\frac{31200}{1.02^{18}}\) - step2: Convert the expressions: \(\frac{31200}{\left(\frac{51}{50}\right)^{18}}\) - step3: Simplify the expression: \(\frac{31200}{\frac{51^{18}}{50^{18}}}\) - step4: Rewrite the expression: \(\frac{31200\times 50^{18}}{51^{18}}\) The present value (PV) is approximately $21,844.97. Now, let's calculate the amount of interest earned: \[ Interest = FV - PV \] \[ Interest = 31,200 - 21,844.97 \] \[ Interest = 9,355.03 \] Therefore, the amount of interest earned is approximately $9,355.03.