You work at a farm that wants to build three identical rectangular corrals in a row against the side of a barn using 112 ft of fencing. No fence is needed next to the barn itself. What dimensions (for each individual corral) will maximize the corral areas

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)  =  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