SOLUTION: A Farmer plans to use 250 feet of fencing to enclose a rectangular corral behind a barn. The back of the barn is 50 feet long and will serve as part of the boundary for the corral.

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Question 406746: A Farmer plans to use 250 feet of fencing to enclose a rectangular corral behind a barn. The back of the barn is 50 feet long and will serve as part of the boundary for the corral. Find the maximum area of corral.
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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A Farmer plans to use 250 feet of fencing to enclose a rectangular corral behind a barn.
The back of the barn is 50 feet long and will serve as part of the boundary for the corral.
Find the maximum area of corral.
:
The perimeter which includes the 50' provided by the barn
L + (L-50) + 2W = 250
2L + 2W = 250 + 50
2L + 2W = 300
Simplify, divide eq by 2
L + W = 150
L = (150-W); use this form for substitution
:
Area
A = L * W
Replace L with (150-W)
A = (150-W) * W
a quadratic equation where area is a function of W
f(W) = -W^2 + 150W
We can find the width for the greatest area by finding the axis of symmetry,
the formula for that is w =-b/(2a); in this equation a=-1; b=150
w = %28-150%29%2F%282%2A-1%29
w = 75 ft is the width for max area
Find L
L = 150 - 75 = 75 also
:
Max Area = 75^2 = 5625 sq/ft
:
Let's see if that works, subtracting the side of the barn
75 +(75-50) + 2(75) =
75 + 25 + 150 = 250 ft of fence