a barn is 150 feet long and 75 feet wide. the owner has 240 ft of fencing and wants to construct two adjacent outdoor pens of identical size along the long side of the barn using the barn as one side of each pen. the owner wants the pens as large as possible. what are the dimensions of the pens that will maximize the area?

Question

Steve · Answer

If the width of the pens is x, and the overall length is y, then

3x+y = 240
The area is

a = xy = x(240-3x) = 240x - 3x^2
This is just a parabola, with vertex at x = 40

So, the pens are each 40 by 60 ft.

As usual, maximum area is achieved when the available fence is divided equally among lengths and widths.