Question 23, 5.2.37 HW Score: $25.81 \%$ Points: 0 of 1 Find the size of each of 9 payments made at the end of each year into a $9 \%$ rate sinking fund which produces $\$ 73000$ at the end of 9 years. The payment size is $\$ \square$. (Round to the nearest cent.)
Solution
Step 1 ：Given that the future value (FV) of the annuity is $73000, the interest rate per period (r) is 9% or 0.09, and the number of periods (n) is 9 years, we need to find the payment made each period (PMT).
Step 2 ：We use the formula for the future value of an ordinary annuity, which is \(FV = PMT \times \frac{(1 + r)^n - 1}{r}\).
Step 3 ：We rearrange the formula to solve for PMT: \(PMT = FV \times \frac{r}{(1 + r)^n - 1}\).
Step 4 ：Substitute the given values into the formula: \(PMT = 73000 \times \frac{0.09}{(1 + 0.09)^9 - 1}\).
Step 5 ：Calculate the right side of the equation: \(PMT = 6570 \div ((1.09)^9 - 1)\).
Step 6 ：Simplify the equation: \(PMT = 6570 \div 0.9476\).
Step 7 ：Finally, calculate PMT: \(PMT = 6933.76\).
Step 8 ：So, the size of each of the 9 payments made at the end of each year into a 9% rate sinking fund which produces $73000 at the end of 9 years is approximately \(\boxed{6933.76}\).
