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# Thinking Outside the Box -- or Maybe Just About the Box

### Introduction

Perhaps one of the deepest entrenched calculus problem is the canonical box problem that students encounter when discussing applied extrema problems. In fact, Friedlander and Wilker (1980) commented, ''This question must be answered nearly a million times a year by calculus students from every corner of the globe'' (p. 282). Without a doubt, almost every recently published calculus text contains a problem similar to the following:

A sheet of cardboard is rectangular, 14 inches long and 8.5 inches wide. Congruent squares are to be cut from its four corners. The resulting piece of cardboard is to be folded and its edges taped to form an open-topped box (see figure below). How should this be done to get a box of largest possible volume?

Figure 1: Canonical Box Problem

In particular, this problem (or some derivative of it) occurs in a classic text such as Thomas (1953) as well as modern texts such as Stewart (2008), Larson, Hostetler and Edwards (2007) and Smith and Minton (2007). But, what about this problem makes it so common and prevalent to the calculus experience? This paper takes a closer look at this common box problem and asks some difficult questions:

• Does this box problem really reflect reality?
• Is there a better box problem more consistent with modern box building techniques? And
• Can a better box problem be explored by Calculus 1 students in a single variable course?

If you enjoy this canonical box problem, we developed a Java Sketchpad applet, called OpenBox, for this problem. For this applet, the yellow points are moveable and control the size of the sheet of cardboard and the position of the cut. A dynamical graphic contained in a grey box shows the graph of the Volume to cut length (x) function. In addition, a dynamical picture of the open box is provided to illustrate how the changes in cut length impacts the configuration of the open box. (Warning: The OpenBox page is best viewed in 1024 x 768 resolution or greater and may take up to a minute to load.)

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Meel, David and Thomas Hern, "Thinking Outside the Box -- or Maybe Just About the Box," Loci (August 2009), DOI: 10.4169/loci003321