Apex Fabricating wants to accumulate $\$ 810,000$ for an expansion expected to begin in four years.
If today Apex makes the first of equal quarterly payments into a fund earning $7.25 \%$ compounded monthly, what should the size of these payments be? (Do not round intermediate calculations and round your final answer to 2 decimal places.)
The quarterly payments

Step 1 ：The problem is asking for the size of equal quarterly payments that Apex Fabricating needs to make in order to accumulate $810,000 in four years. The fund earns an interest of 7.25% compounded monthly.

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

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

Step 4 ：In this case, FV = $810,000, r = 7.25% = 0.0725, n = 12 (since interest is compounded monthly), and t = 4. However, since payments are made quarterly, we need to adjust n and t accordingly. Since there are 4 quarters in a year, n = 4 and t = 4*4 = 16.

Step 5 ：Let's plug these values into the formula and calculate P: \(P = 810000 \times \frac{0.0725/4}{(1 + \frac{0.0725}{4})^{16*4} - 1}\)

Step 6 ：By calculating the above expression, we get P = 6806.421732154582

Step 7 ：Final Answer: The size of the equal quarterly payments that Apex Fabricating needs to make in order to accumulate $810,000 in four years is approximately $6806.42. Therefore, the answer is \(\boxed{6806.42}\)