Answer:
x  =  14 ft
y  =  56 ft
A(max corral)  =  784/3  =  261.33 ft²
Step-by-step explanation:
Let´s call y the side parallel to the barn, then only one y is going to be fenced.
If we are going to divide the area in three identical corrals we need 4 times x ( the other side perpendicular to the barn)
The perimeter of the rectangular area ( divide in three identical corrals)
112  =  y  +  4*x     or   y  =  112  -  4*x
A (r) Â = Â x*y
Area as a function of x  is
A(x) = x* ( 112  -  4*x)     A(x)  =  112*x  -  4*x²
Tacking derivatives on both sides of the equation
A´(x)  =  112  - 8*x    A´(x) = 0   112  -  8*x  =  0
x  =  112/8
x  =  14 ft
And  y  =  112  -  4*x   y  =  112 - 56
y  =  56 ft
A(max)  =  14 * 56  =  784 ft²
A(max corral)  =  784/3  =  261.33 ft²
How do we know the area is maximum, tacking the second derivative
A´´(x)  =  - 8   A´´(x) < 0
Then the function A(x) has a maximum at the point x = 14
