Question

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.

Answer 1

Answer:

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.