You would like to have \( \$ 3,000 \) in 3 years for a special vacation following graduation by making deposits at the end of every six months in an annuity that pays \( 4 \% \) compounded semiannually. a. Determine how much you should deposit at the end of every six months. b. How much of the \( \$ 3,000 \) comes from deposits and how much comes from interest? a. In order to have \( \$ 3,000 \) in 3 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 \( \$ 3,000 \) 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.) be

To solve this problem, we need to use the formula for the future value of an ordinary annuity. The formula is given by: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] Where: - \( FV \) is the future value of the annuity (which we want to be \( 3000 \)) - \( P \) is the amount deposited at the end of every period (which we're solving for) - \( r \) is the interest rate per period - \( n \) is the total number of periods In this case, since the interest is compounded semiannually, the interest rate \( r \) per period will be \( 0.04/2 = 0.02 \), and the total number of periods \( n \) is \( 3 \times 2 = 6 \). Now, we can substitute these values into the formula: \[ 3000 = P \times \frac{(1 + 0.02)^6 - 1}{0.02} \] Calculating \( (1 + 0.02)^6 \): \[ (1.02)^6 \approx 1.12616 \] Now plug this into the equation: \[ 3000 = P \times \frac{1.12616 - 1}{0.02} = P \times \frac{0.12616}{0.02} \approx P \times 6.308 \] Now solve for \( P \): \[ P \approx \frac{3000}{6.308} \approx 475.47 \] Thus, you would need to deposit approximately \( 475.47 \) at the end of every six months. Rounding up, you should deposit \( \$ 476 \). Now, to find out how much comes from deposits and how much from interest: Total deposits over 6 periods: \[ Total\, Deposits = P \times n = 476 \times 6 = 2856 \] Now subtract from the future value to find the interest earned: \[ Interest = 3000 - 2856 = 144 \] Thus, a. In order to have \( \$ 3,000 \) in 3 years, you should deposit \( \$ 476 \) at the end of every six months. b. \( \$ 2,856 \) of the \( \$ 3,000 \) comes from your deposits and \( \$ 144 \) comes from interest.