You would like to have \( \$ 3,500 \) in 4 years for a special vacation following graduation by making deposits at the end every six months in an annuity that pays \( 4 \% \) compounded semiannually. a. Use one of the formulas below to determine how much you should deposit at the end of every six months. \( A=\frac{P\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]}{\left(\frac{r}{n}\right)} \quad P=\frac{A\left(\frac{r}{n}\right)}{\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]} \) b. How much of the \( \$ 3,500 \) comes from deposits and how much comes from interest?

First, let's determine how much you should deposit at the end of every six months. Using the formula \( P = \frac{A\left(\frac{r}{n}\right)}{\left[\left(1+\frac{r}{n}\right)^{nt}-1\right]} \): - \( A = 3500 \), \( r = 0.04 \), \( n = 2 \), and \( t = 4 \). - Therefore, \( P = \frac{3500 \times \left(\frac{0.04}{2}\right)}{\left[\left(1+\frac{0.04}{2}\right)^{2 \times 4}-1\right]} \). - Plugging in these values, we find \( P \approx 393.40 \). So, you'll need to deposit about \( \$393.40 \) every six months. Now, let’s see how much of the \( \$3,500 \) comes from your deposits versus how much comes from interest. Over 8 deposits (2 per year over 4 years): - Total deposits: \( 393.40 \times 8 = 3,147.20 \). - Interest earned: \( 3,500 - 3,147.20 = 352.80 \). So, from this, your deposits contribute \( \$3,147.20 \) while interest adds another \( \$352.80 \). Happy savings for that vacation!