Answer: Max area = 640000 square yards
The rectangle is 800 yards by 800 yards (i.e. this rectangle is a square).
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Work Shown:
L = length
W = width
P = 2(L+W) = perimeter of the rectangle
3200 = 2(L+W)
3200/2 = L+W
1600 = L+W
W = 1600-L
A = area of the rectangle
A = L*W
A = L*(1600-L)
A = 1600L - L^2
A = -L^2 + 1600L
Replace L with x to get the area expression of -x^2+1600x
If you were to graph this and look for the vertex, then you should see it's located at (800, 640000) which is the highest point of this parabola.
That tells us the largest area value y = 640000 occurs when the length is x = 800
Then notice W = 1600-L = 1600-800 = 800
This rectangle is really a square with side length 800 by 800.
It's no coincidence that we end up with a square. It turns out that if you want the most area for a given fixed perimeter of some rectangle, then a square is the best choice.