Maximum Area of the Tennis Court

Question

A fenced tennis court is being built right next to an existing fenced tennis court so that the new court will only need three new sides of fencing built. The builder has 240ft of fencing to use on these three sides. What is the maximum area of the new tennis court?

Answer 1

Let the new court be a rectangle L ft by W ft, with W being the side that has existing fencing.

The perimeter needing fencing is then 2L + W = 240. Solving for W in terms of L gives W = 240 - 2L.

The area is A = LW and plugging in the expression above for W gives A = L (240 - 2L) = -2L² + 240L

This area function is just a concave down parabola with a vertex (maximum) at L = -b/2a = -240/2(-2) = 60

So L = 60 ft, W = 120 ft, and the maximum area is 7,200 ft².

Notice the symmetry of this answer, using 120 for the width and a total of 120 for the lengths. Notice we make W bigger because one side of W is already fenced. If the existing fence wasn't there, you can use the same process to show that a 60' by 60' square would maximize the enclosed area under the constraint of having only 240' of perimeter to use.

(For a rectangle, that is. If we're allowed to use 240' of perimeter to make any shape, then we would construct a circle with circumference = 240, r = 120/π , and A = 14,400/π which is about 25% better than the 3,600 we got from the square.)