Carol has 800 ft of fencing to fence in a rectangular horse corral which is bordered by a barn on one side. find the dimensions that maximizes the area of the corral.

200 ft * 400 ft

Step-by-step explanation:

Let x = one dimension of the corral

and y = the other dimension

Carol is using the barn on one side, so she needs to fence in only three sides (as in the diagram below).

Then

(1) 2x + y = 800 (Formula for perimeter)

(2) y = 800 - 2x Subtracted 2x from each side

(3) A = xy (Formula for area)

A = x (800 - 2x) Substituted (2) into (3)

A = 800x - 2x² Distributed the x

This is the equation for a downward-opening parabola.

The vertex (maximum) occurs at

x = - b / (2a), where

b = 800 and

a = - 2

x = - 800/[2 (-2) ] = - 800 / (-4)

(4) x = 200 ft

This is the value of x that gives the maximum area.

2*200 + y = 800 Substituted (4) into (1)

400 + y = 800

y = 400 ft Subtracted 400 from each side

This is the value of y that gives the maximum area.

The dimensions that maximize the area of the corral are 200 ft * 400 ft.

The graph of A = 800x - 2x² shows that the area is a maximum

when x = 200 ft.