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.
