A Sheep-pen Problem

The problem is:

A farmer has 60 hurdles, with which she plans to build a sheep-pen. She wants the area of the pen to be as large as possible. Two hurdles can be fixed together end-to-end to form a straight line, or to be at right angles; no other angles are possible.

Where she plans to build the pen, there's a long straight sheep-proof fence. If the end of a hurdle is next to the fence, she can make a sheep-proof join. She realises that she can use the fence to form one side of the pen.

What shape should she make the pen?

You may want to try to solve the problem, before you look at the two solutions below.

Solution using calculus: show

Solution using symmetry: show

Comparison of solutions: show