The Pen and the Barn

Abstract

A standard differential calculus problem involves showing that the rectangle with fixed perimeter and maximal area is a square. Textbooks often present this in terms of a farmer who wishes to build the most efficient rectangular pen with a fixed length of fencing. A variant uses one wall of a barn for all or part of one side of the pen (Figure 1). Let's take a new look at this latter class of problems, because there is an attractive way to show how the lengths of the fence and the barn wall affect the shape of the optimal pen.