You would like to have \( \$ 2,500 \) in 5 years for a special vacation following graduation by making deposits at the end of every six months in an annuity that pays \( 4.5 \% \) compounded semiannually. a. Determine how much you should deposit at the end of every six months. b. How much of the \( \$ 2,500 \) comes from deposits and how much comes from interest? a. In order to have \( \$ 2,500 \) in 5 years, you should deposit \( \$ \square \) at the end of every six months. (Do not round until the final answer. Then round up to the nearest dollar.) b. \( \$ \square \) of the \( \$ 2,500 \) comes from your deposits and \( \$ \square \) comes from interest. (Use the answer from part a to find this answer. Round to the nearest dollar as needed.)

To determine how much you should deposit at the end of every six months in an annuity that pays \( 4.5\% \) compounded semiannually, we can use the future value of an annuity formula: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] where: - \( FV \) is the future value (\$2,500), - \( P \) is the periodic deposit, - \( r \) is the interest rate per period, - \( n \) is the total number of deposits. Given that the annual interest rate is \( 4.5\% \) compounded semiannually, the interest rate per period \( r \) is \( \frac{0.045}{2} = 0.0225\). Over 5 years, you would make \( 5 \times 2 = 10 \) deposits. Now we can rearrange the formula to solve for \( P \): \[ 2500 = P \times \frac{(1 + 0.0225)^{10} - 1}{0.0225} \] Calculating \( (1 + 0.0225)^{10} \): \[ (1 + 0.0225)^{10} \approx 1.24568 \] Thus: \[ 2500 = P \times \frac{1.24568 - 1}{0.0225} \] \[ 2500 = P \times \frac{0.24568}{0.0225} \approx P \times 10.9078 \] Now solve for \( P \): \[ P \approx \frac{2500}{10.9078} \approx 229.73 \] Rounding up, you should deposit approximately \( \$230 \) at the end of every six months. Next, we can figure out how much of the \$2,500 comes from deposits and how much comes from interest. The total amount deposited over 5 years (i.e., 10 deposits) would be: \[ \text{Total Deposits} = P \times n = 230 \times 10 = 2300 \] To find the interest earned, we subtract the total deposits from the future value: \[ \text{Interest Earned} = FV - \text{Total Deposits} = 2500 - 2300 = 200 \] So, in short: a. In order to have \( \$2,500 \) in 5 years, you should deposit \( \$230 \) at the end of every six months. b. \( \$2,300 \) of the \( \$2,500 \) comes from your deposits and \( \$200 \) comes from interest.