Question

A rectangular pen is built with one side against a barn. two hundred meters of fencing are used for the other three sides of the pen. what dimensions maximize the area of the pen?

Answer 1

Let's call the repeated side of the barn
`x`. Then the remaining fence for the other side is
`200-2x` so the area is

`A = x(200-2x) = -2x^2+200x = -2(x^2-100x)=-2(x^2-100x+2500)- -2(2500) = -2(x-50)^2+5000`

That clearly has a maximum at x=50, so 50 by 100 for the maximal area.