The total sales, $S$, of a one-product firm are given by $S(L, M)=M L-L^{2}$, where $M$ is the cost of materials and $L$ is the cost of labor. Find the maximum value of this function subject to the budget constraint shown below.
\[
M+L=125
\]
The maximum value of the sales is $\$$ (Round to the nearest cent as needed.)

Step 1 ：The problem is a constrained optimization problem. We need to find the maximum value of the function \(S(L, M)=M L-L^{2}\) subject to the constraint \(M+L=125\).

Step 2 ：To solve this problem, we can substitute \(M\) from the constraint into the function \(S(L, M)\), and then find the derivative of the resulting function with respect to \(L\).

Step 3 ：Setting the derivative equal to zero will give us the value of \(L\) that maximizes the function.

Step 4 ：We can then substitute this value of \(L\) back into the function to find the maximum value.

Step 5 ：Let's start by substituting \(M\) from the constraint into the function \(S(L, M)\). We get \(S = -L^{2} + L*(125 - L)\).

Step 6 ：Next, we find the derivative of \(S\) with respect to \(L\). We get \(S' = 125 - 4*L\).

Step 7 ：Setting the derivative equal to zero, we get \(L_{max} = \frac{125}{4}\).

Step 8 ：Substituting this value of \(L_{max}\) back into the function, we get \(S_{max} = \frac{15625}{8}\).

Step 9 ：Final Answer: The maximum value of the sales is \(\boxed{\$1953.13}\).