Optimization for Ranchers

Premise: a rancher wants to build a rectangular lattice of pens to house livestock. Given that the fencing will be laid out in the sort of pattern illustrated below, find the optimal dimensions for the rectangle. Either you are trying to surround a lot of area with a fixed amount of fencing, or you're trying to surround a fixed amount of area with the smallest possible amount of fence. These are classic optimization problems--usually approached via calculus--but algebra can also solve them (such as by completing the square). This tool will help you visualize how changing the dimensions of the rectangle (while obeying some constraint) will affect the quantity you are trying to optimize.