## Maximum Volume & Area with the Geometer's Sketchpad

### by David Purdy

Common in calculus and advanced algebra texts for years, these well known optimization problems now appear in several high school geometry texts, as interesting explorations that integrate measurement, model building, polynomial functions and use of the graphing calculator. The sketches listed below provide another facet through which to dynamically view and explore this rich genre of problems.

1. A manufacturer of paper products wishes to construct the largest box possible from a 10-inch-square piece of flat paper. The box is to be made by cutting out congruent squares from each of the four corners. How big should the cut-out squares be, and what should the dimensions of the box be in order to maximize its volume? How much will this optimum box hold? What fractional part of the side length should the cut-out be in order to maximize the box's volume?

2. The same situation, except that the sheet metal is rectangular, rather than square.

3. A farmer wishes to fence in the largest possible area, using only 20 linear meters of fencing material for three sides of the pen, and the barn for the fourth side. How much area can be fenced in this way, and what are the optimum dimensions of the pen?

Created and submitted by David Purdy, Wisconsin Heights High School, 10173 Highway 14, Mazomanie WI 53528. The author wishes to thank the National Science Foundation and its support through Grant No. ESI: 95-54095, and the NCTM for its support of the Presidential Award for Excellence in Mathematics Teaching program. Date: 6/5/98