In lots of problems, the problem can be simplified or even solved by exploiting symmetries in the problem.

Example 1

A farmer needs to construct an animal pen in the shape of a rectangle. He has enough materials to construct a \(1000\) meters of fencing.

Problem: What are the unique dimensions of the pen that maximize the area enclosed?

To solve this problem, let \(x\) be the width of the pen and \(y\) be the length. Then the area that we are trying to optimize is \(A = xy\). Notice that if \(x = a\) and \(y=b\) maximize the area, then so do \(x = b\) and \(y=a\). Since the optimal dimensions are unique, it follows that \(a = b\).

The optimal dimensions will use up all the supplies, so \(2a + 2b = 1000\). Since \(a = b\), this means \(2a + 2a = 1000\), ie. \(4a = 1000\) so that \(a = 250\). Consequently the optimal dimensions are a width of \(250\) and a length of \(250\).

Note: It is not always true that when we are trying to optimize a symmetric function \(f(x,y)\) that the optimum will occur at a symmetric point. Can you think of a counter-example?