Determine the present value $P$ that must be invested to have the future value $A$ at simple interest rate $r$ after time $t$. \[ A=\$ 4000.00, r=10.5 \%, t=9 \text { months } \] $\$ \square$ (Do not round until the final answer. Then round up to the nearest cent as needed.)

Step 1 ：We are given the future value of the investment (\(A = \$4000.00\)), the annual interest rate (\(r = 10.5\%\)), and the time the money is invested for (\(t = 9\) months). We are asked to find the present value (\(P\)) that must be invested.

Step 2 ：The formula for the future value of an investment using simple interest is \(A = P(1 + rt)\). We can rearrange this formula to solve for \(P\): \(P = \frac{A}{1 + rt}\).

Step 3 ：Before we can use this formula, we need to convert the interest rate from a percentage to a decimal by dividing by 100, and convert the time from months to years by dividing by 12. This gives us \(r = 0.105\) and \(t = 0.75\).

Step 4 ：Substituting these values into the formula gives us \(P = \frac{4000.0}{1 + 0.105 \times 0.75}\).

Step 5 ：Calculating this gives us \(P = 3707.9953650057932\).

Step 6 ：Rounding this to the nearest cent gives us \(P = \$3707.99\).

Step 7 ：Final Answer: The present value that must be invested to have the future value of \$4000.00 at a simple interest rate of 10.5% after 9 months is approximately \(\boxed{3707.99}\).