Horse Breeder Wants to Construct

Question

A horse breeder wants to construct a corral next to a horse barn that is L=20 feet long, using the barn as part of one side of the corral as shown in the figure above. The breeder has 360 feet of fencing available.
Find the value of x
which maximizes the amount of area the corral can enclose:

Answer 1

Answer is 95 ft by 95 ft by adding 75 ft to the 20 ft of length of the barn. 

If you only use the barn only as one side, you can have sides of 20 & 170

which gives an area of 3400 sq ft.

But if you add a length 'y' to the barn and run a width of 'x'

Area = ( 20 + y ) x  =  xy + 20x

Perimeter excluding the barn = 360 = 2x + 2y + 20

Solving for y =>  

y = 170 - x

Using substitution to eliminate 'y'

A =  x ( 170 - x ) + 20x

Expanding and then differentiating

dA/dx = - 2x + 190

Setting this to zero yeilds

x = 95

and thus y = 170 - x = 75

So the maximum area A = 95 x 95 = 9025

 

Also since A''. the 2nd derivative is = -2

being negative tells us this is a local maximum

rather than a local minimum