Let's denote the length of the playground by L and the width by W. We want to find the dimensions that will maximize the total area of the playground, given that 720 feet of fencing is used.
We can start by using the fact that the total length of the fencing is 720 feet:
2L + W = 720
We can then solve this equation for W in terms of L:
W = 720 - 2L
The total area of the playground is given by:
A = LW
Substituting the expression for W into the area equation, we get:
A = L(720 - 2L)
Expanding and simplifying, we get:
A = 720L - 2L^2
To find the dimensions that maximize the area, we can take the derivative of A with respect to L, set it equal to zero, and solve for L:
dA/dL = 720 - 4L = 0
Solving for L, we get:
L = 180
Substituting this value of L back into the equation for W, we get:
W = 720 - 2(180) = 360
Therefore, the dimensions of the playground that will enclose the greatest total area are 180 feet by 360 feet.