Find the amount of each payment to be made into a sinking fund eaming $8 \%$ compounded monthly to accumulate $\$ 49,000$ over 5 years. Payments are made at the end of each period.

The payment size is $\$ \square$.
(Do not round until the final answer. Then round to the nearest cent.)
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Clieck Answer

Step 1 ：We are given a future value of the annuity as $49,000, an annual interest rate of 8% or 0.08, the interest is compounded 12 times a year, and the time period is 5 years. We are asked to find the payment made each period.

Step 2 ：We use the formula for the future value of an ordinary annuity: \(FV = P \times \frac{(1 + \frac{r}{n})^{nt} - 1}{\frac{r}{n}}\)

Step 3 ：We rearrange this formula to solve for P: \(P = FV \times \frac{\frac{r}{n}}{(1 + \frac{r}{n})^{nt} - 1}\)

Step 4 ：Substitute the given values into the formula: \(P = 49000 \times \frac{\frac{0.08}{12}}{(1 + \frac{0.08}{12})^{12 \times 5} - 1}\)

Step 5 ：Solving the above expression, we get \(P = 666.876653465612\)

Step 6 ：Rounding to the nearest cent, we get the final answer: \(\boxed{\$666.88}\)