Question

Gina has 1200 yards of fencing to enclose a rectangular area. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum​ area?

Answer 1

90,000 square yards

Answer 2

We know the perimeter formula would be 1200=2x+2y. And the area will be A=x*y
We can to take the derivative of the area formula to find where it is a max but we must first substitute something in from the first formula so we only have one variable.

1200-2x=2y
600-x=y

A=(x)*(600-x)
A=600x-x^2

Now we take the derivative:
A'=600-2x (set equal to 0 and solve)
0=600-2x
2x=600
x=300

Then when we plug this into the perimeter formula, we can solve for y
1200=2x+2y
1200=2(300)+2y
1200=600+2y
600=2y
y=300

So both the length and width are 300 yards, and the area would be 90,000 square yards

Hope this helps