Question

One side of a rectangular animal pen is to be formed by the side of a barn. If you have enough material to build a total of 34 feet of fence, find the maximum area than can be enclosed

Answer 1

Let side x be parallel to the barn. Then

x+2y = 34
a = xy = (34-2y)y = 34y-2y^2
da/dy = 34-4y
max area when da/dy=0, or y=17/2

So, the pen is 17 by 17/2

As usual, max area is when the fencing is divided equally among lengths and widths.