Question

Suppose that you have 1600 m of fencing with which to build three adjacent rectangular corrals, as shown in the figure. Find the dimensions so that the total enclosed area is as large as possible.

Answer 1

as shown in which figure?

Answer 2

I don't see an image, so I'll assume that they are set up as I've drawn them.

The total area will be equal to the height times the width. The length of fencing will be equal to 4h + 2w.
4h + 2w = 1600
A = hw
Make area in terms of just one variable:
\[w = 800 - 2h\]
\[A = h(800 - 2h)\]
\[A = 800h - 2h^{2}\]
Maximize Area by finding A', setting it equal to zero and finding a turning point:
\[A' = 800 - 4h\]
\[0 = 800 - 4h\]
\[4h = 800\]
\[h = 200\]

So h = 200. w = 400. You can see that 4h + 2w = 1600. The maximum area is 400*200.