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.
:
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 = {{{(-150)/(2*-1)}}}
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