That Stingy Rancher And His Fence Again

Geometry Level 5

A rancher has a square mile of flat land, a perfect square with a fenced perimeter of 4 miles. He decides he wants to divide his property into 4 parts of equal areas. He knows he can do this with 2 miles of fencing, which is $1000 per mile, but he is feeling particularly stingy and thinks he can do it with less. Let A = the total length of fence. A is minimized. How much did he pay for fencing, expressed as \(\left\lfloor 1000A \right\rfloor \)?

This problem can be done without calculus. The 4 parts need not be of the same shape. The answer is less than \(2000\).

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