Question

A rectangular playground is to be fenced off and divided into two parts by a fence parallel to one side of the playground. 240 feet of fencing is used. find the dimensions of the playground that will enclose the greatest total area.

Answer 1

Let's look at the picture, let's imagine that the gray line is the perimeter fence and that the red OR the blue is the one dividing it. We can see that the blue line is longer than the red one, so it will be advantageous, to have a bigger area, to have the dividing fence the smallest possible.
Let's say then that the width (W) is bigger (or equal) to the length (L), so we have:
$W \geq L W+W+L+L+L=240 2W+3L=240 W=\frac{240-3L}{2}$
The area is W*L, so we have
$A=W*L=L*\frac{240-3L}{2}=\frac{-3}{2}(L^2-80L)=\frac{-3}{2}L(L-80)$
this function is a parabola facing down, its zeros are 0 and 80, therefore its maximum is when L=40
hence, L=40 and W=(240-120)/2=60
It will be a rectangle, measuring 60x40 and the divinding fence will be 40