Step 1 :Let's denote the length of the rectangle as \(x\) and the width as \(y\). Since the playground is divided into two by another fence parallel to one side, we have \(2x + 3y = 768\). We need to find the maximum of the area \(A = xy\).
Step 2 :We can express \(y\) in terms of \(x\) from the equation of the perimeter, and then substitute it into the equation of the area. This will give us a function of one variable that we can maximize.
Step 3 :Express \(y\) in terms of \(x\) as follows: \(y = 256 - \frac{2x}{3}\)
Step 4 :Substitute \(y\) into the area equation to get: \(A = x*(256 - \frac{2x}{3})\)
Step 5 :Take the derivative of \(A\) with respect to \(x\) to get: \(A' = 256 - \frac{4x}{3}\)
Step 6 :Set \(A'\) equal to zero and solve for \(x\) to find the critical points: \(x = 192\)
Step 7 :Substitute \(x = 192\) back into the equation \(y = 256 - \frac{2x}{3}\) to find \(y = 128\)
Step 8 :Substitute \(x = 192\) and \(y = 128\) into the area equation to find the maximum area: \(A = 24576\)
Step 9 :Final Answer: The maximum area of the playground is \(\boxed{24576}\) square feet.