Find the amount of each payment to be made into a sinking fund earning $9 \%$ compounded monthly to accumulate $\$ 71,000$ over 8 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.)

Step 1 ：We are given that the sinking fund earns 9% interest compounded monthly and we want to accumulate $71,000 over 8 years. We are asked to find the amount of each payment to be made into the fund. Payments are made at the end of each period.

Step 2 ：The formula for the future value of a series of payments (an annuity) is: \(FV = P \times \frac{(1 + \frac{r}{n})^{nt} - 1}{\frac{r}{n}}\) where: FV is the future value of the annuity (in this case, $71,000), P is the payment made each period (this is what we're trying to find), r is the annual interest rate (9% or 0.09 in decimal form), n is the number of times the interest is compounded per year (12 times for monthly compounding), t is the number of years the money is invested for (8 years).

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

Step 4 ：Let's plug in the given values and calculate P: FV = 71000, r = 0.09, n = 12, t = 8

Step 5 ：By calculating, we get P = 507.6644324321948

Step 6 ：Rounding to the nearest cent, we get the final answer: \(\boxed{507.66}\)