Answer
(a) The periodic deposit is $\$442$
(b) The total amount of money deposited into the annuity is $\$3536$
The interest is $\$464$
Work Step by Step
(a) This is the formula we use to calculate the value of an annuity:
$A = \frac{P~[(1+\frac{r}{n})^{nt}-1]}{\frac{r}{n}}$
$A$ is the future value of the annuity
$P$ is the amount of the periodic deposit
$r$ is the interest rate
$n$ is the number of times per year the interest is compounded
$t$ is the number of years
$A = \frac{P~[(1+\frac{r}{n})^{nt}~-1]}{\frac{r}{n}}$
$P = \frac{A~(\frac{r}{n})}{~(1+\frac{r}{n})^{nt}~-1}$
$P = \frac{(\$4000)~(\frac{0.07}{2})}{~(1+\frac{0.07}{2})^{(2)(4)}~-1}$
$P = \$442$
The periodic deposit is $\$442$
(b) The total amount of money deposited into the annuity is $\$442 \times 8$, which is $\$3536$
The interest is the difference between the value of the annuity and the total amount deposited. We can calculate the interest.
$interest = \$4000 - \$3536 = \$464$
The interest is $\$464$
