A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. 768 feet of fencing is used. Find the maximum area of the playground.

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.