Well lets have x be the side perpendicular to the barn. You will have two sides of length "x". Which means the side parallel to the barn has the length of (200 -2x)
So we know the area of the pasture is length * width or x * (200 - 2x)
This means we are seeking to maximize x * (200 - 2x).
This is actually a parabola with zeroes that are at x = 0 and x = 100 which means the vertex is at x = 50.
So when x = 50 > (200 - 2x) = 100
So that means the maximum area of the pasture is 50 * 100 = 5000 square feet.
