Question 5 Notyetanswered Points out of 10.00 p Flag question Jon wishes to accumulate $$ \$ 4,500 $$ in 8 years for a long vacation. Use Table $$ 12-1 $$ to find the sinking fund payment he would need to make at the END of every six months, at $$ 4 \% $$ interest compounded semiannually. Select one: a. $$ \$ 241,43 $$ b. $$ \$ 233.97 $$ c. $$ \$ 207.63 $$ d. $$ \$ 344.10 $$

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Let the sinking fund accumulation factor for a level payment at the end of each period be

$$
a_{\overline{n}\,|\,i} = \frac{(1+i)^{n} - 1}{i}
$$

where $$ i $$ is the interest rate per period and $$ n $$ is the total number of periods.

Since the interest rate is $$ 4\% $$ compounded semiannually, the interest rate per six-month period is

$$
i = \frac{0.04}{2} = 0.02
$$

and the number of six-month periods in 8 years is

$$
n = 8 \times 2 = 16.
$$

The sinking fund equation is

$$
\text{Payment} \times a_{\overline{16}\,|\,0.02} = \$4500.
$$

First, compute the accumulation factor:

$$
a_{\overline{16}\,|\,0.02} = \frac{(1.02)^{16} - 1}{0.02}.
$$

Approximating $$(1.02)^{16}$$:

$$
(1.02)^{16} \approx 1.373
$$

thus

$$
a_{\overline{16}\,|\,0.02} \approx \frac{1.373 - 1}{0.02} = \frac{0.373}{0.02} \approx 18.65.
$$

Now, solve for the sinking fund payment:

$$
\text{Payment} = \frac{4500}{18.65} \approx 241.43.
$$

Therefore, the required sinking fund payment is approximately $$ \$241.43 $$.

The correct answer is:

a. $$ \$241.43 $$
The sinking fund payment Jon needs to make is approximately \$241.43 every six months.

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