A rancher has 4000 ft. of fence for constructing a rectangular corral that is divided into 3 equal sections. one side of the corral will be bounded by his barn and needs no fencing. what are the dimensions of the corral that will maximize the area? what is the area of the corral?

Let x and y be the sides of the rectangular corral. Therefore,

Area, A = x*y = xy

Additionally, let the side bordering the barn be x. Then,

Circumference, C = x+y+y = x+2y = 4000 ft = > x = 4000 - 2y

Substituting for x in the area equation,

A = (4000-2y) y = 4000y - 2y^2

For maximum area, dA/dy = 0

Then,

dA/dy = 4000 - 4y = 0 = > 4000 = 4y = > y = 4000/4 = 1000 ft

And x = 4000 - 2y = 4000 - 2*1000 = 4000 - 2000 = 2000 ft

The dimensions of the corral are: 2000 ft by 1000 ft.

Area, A = xy = 2000*1000 = 2000,000 ft^2