A rectangular playground of sides a and b would have an area of a*b. The perimeter of the fence plus one side is 648ft; this can be written as 3a+2b. We can write b in the expression for the area as (648-3a)/2. The area is equal to (648a-3a^2)/2; the maximum value can be found by deriving the area expression and equate it to 0.
The derivative is 324-3a; the value of the area is maximum when a=108ft.
The dimensions that maximize the playground are a=108ft and b=162ft.
