You would like to have \( \$ 3,500 \) in 3 years for a special vacation following graduation by making deposits at the end of every six months in an annuity that pays \( 6.5 \% \) compounded semiannually. a. Determine how much you should deposit at the end of every six months. b. How much of the \( \$ 3,500 \) comes from deposits and how much comes from interest?

To determine how much you should deposit at the end of every six months, we can use the future value of an annuity formula: \[ FV = PMT \times \frac{(1 + r)^n - 1}{r} \] Where: - \( FV \) is the future value (in this case, \( \$ 3,500 \)) - \( PMT \) is the amount deposited each period - \( r \) is the interest rate per period - \( n \) is the total number of deposits With a 6.5% annual interest rate compounded semiannually, the semiannual interest rate \( r \) is \( 6.5\%/2 = 3.25\% = 0.0325 \). Over 3 years, you'll make \( n = 3 \times 2 = 6 \) deposits. Rearranging the formula to solve for \( PMT \): \[ PMT = \frac{FV \times r}{(1 + r)^n - 1} \] Now plug in the numbers: \[ PMT = \frac{3500 \times 0.0325}{(1 + 0.0325)^6 - 1} \approx \frac{113.75}{0.2071} \approx 549.13 \] So, you should deposit approximately \( \$ 549.13 \) at the end of every six months. To determine how much of the \( \$ 3,500 \) comes from deposits and how much comes from interest, calculate total deposits and then subtract from future value. Total deposits = \( PMT \times n = 549.13 \times 6 \approx 3,294.78 \). Now calculate the interest earned: Interest = \( FV - \text{Total deposits} \approx 3,500 - 3,294.78 \approx 205.22 \). Thus, from the \( \$ 3,500 \), approximately \( \$ 3,294.78 \) comes from deposits and \( \$ 205.22 \) from interest.