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

### Abstract

We present a fresh look at an age-old calculus problem, the box problem. The box problem is traditionally used as one of the typical exemplars for the study of optimization. However, this paper illustrates that another box problem, tied much closer to reality, can be used to investigate optimization. In particular, we present an improved box problem and its real-world context along with activities that blend hands-on and applet-based investigations with a strong dose of analysis that culminates in other possible extensions and investigations.

#### Appendix: Student Module

(Condensed from the article for student and classroom exploration)

### 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.)

### Background

#### Historical References

The Box problem is so timeless that it even goes back almost a hundred years to the classic Granville and Smith (1911) text. Except for the fact that it is made out of tin, doesn't this look familiar?

Figure 2. The box problem on page 115 in Granville and Smith (1911)

However, this instance was not the first appearance of the box problem in either an academic publication or in popular culture. A derivation of the box problem first appeared in popular culture in the 1903 Henry Dudeny's puzzle column in the Weekly Dispatch and again in 1908, it was rephrased for his puzzle column in Cassell's Magazine. Below is the graphic and phrasing of the problem contained in Cassell's Magazine:

Figure 3. Illustration from Cassell's Magazine.

''No. 525 -- HOW TO MAKE CISTERNS. Here is a little puzzle that will elucidate a point of considerable importance to cistern makers, ironmongers, plumbers, cardboard-box makers, and the public generally.

Our friend the cistern-maker has an interesting task before him. He has a large sheet of zinc, measuring eight feet by three feet, and he proposes to cut out square pieces from the four corners (all, of course, of the same size), then fold up the sides, join them with solder, and make a cistern.

So far, the work appears to be pretty obvious and easy. But the point that puzzles him is this: What is the exact size for the square pieces that he must cut out if the cistern is to contain the greatest possible quantity of water?

Call the feet inches, and take a piece of cardboard or paper eight inches long and three inches wide. By experimenting with this you will soon see that a great deal depends on the size of those squares. To get the greatest contents you have to avoid cutting those squares too small on the one hand and too large on the other. How are you going to get at the right dimensions? I shall award our weekly half-guinea prize for a correct answer. State the dimensions of the squares and try to find a rule that the intelligent working man may understand.''

(Dudeney, 1908, p. 430)

But just in case one might think that the problem is only a century old, a Treatise on Differential Calculus by Todhun

ter (1855) contains the following related problem:

Figure 4. The box problem on page 213 of Todhunter (1855).

Consequently, this problem has appeared in calculus texts for at least the last one and a half centuries. It has become so entrenched in the calculus curriculum and articles are still being written about it. For instance, an article by Miller and Shaw (2007) breathed new life into this problem for it served as the backdrop to explore a conventional problem in unconventional ways or an article by Fredrickson (2003) reconceptualizing the constraints on the ''cutouts'' while striving for a container of maximum volume. But what is it about this problem that makes it so universal and mathematics teachers keep coming back to it? Does it really motivate student learning of essential calculus concepts? We concur with Underwood Dudley (2002) when he pointed out in his talk on Calculus Books, "when have you encountered a cardboard box constructed in such a way?" and the problem is ''quite silly'' according to Friedlander and Wilker (1980), Dundas (1984) and Pirich (1996) since the corners are wasted.

As previously identified, this particular calculus problem predates modern use of cardboard packaging. But, most would agree that constructing a cardboard box in this manner is quite unusual. So, why do calculus teachers keep including it as part of their repertoire of examples other than the fact that it is straightforward and easily comprehensible? Doesn't the inclusion make students begin to wonder about the ability of calculus to model reality? In fact, if one attempts to rip apart the classic open box used for the typical shirt box that one receives at Christmas, it is not constructed in this manner.

It is interesting to note that both the Granville and Smith text and Dudeney's puzzle used tin, a little more practical material than cardboard for such a construction. A box of cardboard might use the squares as tabs to connect the sides together and increase stability and strength. In fact, we have often asked students to construct boxes from index cards we gave them and they tend to do use these tabs naturally to make the connections with tape, glue or a staple. The students can typically answer the mathematical questions of the corner-removed box problem but at the same time they recognize the lack of realism when the corners are removed since boxes just are not constructed that way.

#### Introducing the Regular Slotted Container

It is our contention that there is a better box problem to be examined - a box problem that students will see as realistic as well as to be able to explore with tools from single-variable calculus. This adventure began a little over 25 years ago when one of the authors was curious about how to model realistic shipping boxes and tore one apart to find out. His first exploration was with a box known in the shipping industry as a Regular Slotted Container (RSC). The RSC is perhaps one of the most commonly utilized class of boxes in the shipping industry because it is one of the most economical boxes to manufacture and adapt for the shipment of most commodities. At first exploration, the construction of a RSC appears to require two variables but a simple but practical constraint makes it a problem appropriate for single variable calculus.

Let's first explain what exactly a RSC is. As shown below in figures 5 and 6, a Regular Slotted Container is constructed from a large rectangular sheet of corrugated cardboard. Depending on the construction details, it may be manufactured with a tab used to either glue or stitch the joint (the inclusion of the tab results in waste cardboard during production) or may rely on a taped joint making it a minimal shipping container (i.e. a box where there is ''no'' scrap corrugated cardboard generated in the manufacturing process). In either case, the lengthwise (normally outer) flaps meet at the center of the box allowing it to be affixed by tape or staples.

Figure 5: Construction schematics for the RSC (Safeway Packaging, n.d.)

 The box is closed Top flaps opened All flaps opened Completely disassembled

Figure 6. Various stages of a RSC being disassembled

#### Movie of the construction of the RSC

It is difficult to conceive of how a blank sheet of cardboard with cuts at appropriate points and to particular lengths is transformed into a box that is commonly used across the shipping industry. In order to get a clear sense of this, we provide a link to a movie that shows the SWF Companies (n.d.) CE-451 manufacturing a Regular Slotted Container.

Warning: The CE-451 movie is 12 MB and may take more than a minute to load.

What one sees in this movie are the blanks being loaded into a hopper and the robotic elements of the CE-451 taking those blanks and converting them into RSCs. What it does not show is how the cardboard was sliced, the positioning of those slices and how the slice lengths help maximize the volume capacity of the RSC.

### Discussing Dundas's (1984) analysis

In a paper by Dundas (1984), he looked at a variety of box problems and one of them was configured like the RSC. The tactic that Dundas (1984) took was to look at the problem from the perspective of a piece of cardboard of fixed area rather than a fixed dimensions. We find this choice mathematically sound but problematic to the exploration of the real-world situation since cardboard is not typically sold by area but rather constrained by manufacturable lengths and widths. Consequently, when Dundas (1984) explored the function,

$V(l) = T\left( {{1 \over 2} - T} \right)\left( {{A \over l} - T} \right)$

there was no way to convey to the reader information about the length and width of the cardboard (since the cardboard area was being restricted to A square inches). In addition, a critical piece of the problem is what the lengths of the cuts should be, since that transforms the problem from a multiple-variable problem to a single-variable problem. Unfortunately for a student reading the paper, it was never motivated or explained. It is these particular elements that we wish to revisit and illustrate using text and a set of applet-based activities to lead students to similar conclusions without making the activity all about pushing algebra around on a page.

### The First Box Problem

#### Introduction

Being able to step back and consider the problem without many constraints allows one to see the interdependence of the cut length, positioning of cuts, and the volume of the box. However, will students recognize that if they encounter the following box problem?

A sheet of cardboard is rectangular, 14 inches long and 8.5 inches wide. If six congruent cuts (denoted in black) are to be made into the cardboard and five folds (denoted by dotted lines) made to adjoin the cuts so that the resulting piece of cardboard is to be folded to form a closeable rectangular box (see figures below). How should this be done to get a box of largest possible volume?

Figure 7: The box problem graphics

Before interacting with the Box Problem applet, students should be given the opportunity to physically build their own boxes out of paper or cardboard using scissors and tape or dissect a provided RSC. In doing so, students' physical constructions or destructions will help them recognize the particular physical constraints and how elements of the RSC come together prior to exploring the virtual world presented in the applets. Questions that can be asked of the students during this initial phase of exploration, if they don't naturally generate them on their own, are:

1. What does a ''closeable rectangular box'' infer about the characteristics of the box?
2. What impact does "closeable" have with respect to the length of at least a pair of the flaps?
3. Should the flaps overlap? and
4. If they do overlap, is that the best use of the cardboard?

After students have constructed boxes in this manner then we suggest that students begin to interact with a set of Java applets designed to help students interact with the problem in ways that would not be possible when building boxes out of cardboard.

But where does one go from there? Is moving expressly toward the algebraic solution the only path? It is our contention that considerably more learning can occur if this path is at least delayed a little while longer so students can be asked to explore the problem with greater depth. The pertinent question here is how does one encourage students to search for deeper insights that can be drawn from such a seemingly simple problem. We suggest the use of a pair of applets to help students investigate the problem in increasingly more complex ways and open their eyes to conditions and constraints that were not obvious through their initial encounters.

#### The first Box Problem applet

One element in the applets that we have designed are the multiple representations and alternative ways of conveying information. For instance, the image shown below is a screen shot from the ClosedBox applet, which allows students to explore various scenarios. The student can manipulate the lengths A and B by just pulling on the yellow points A' and B'. The yellow points P and Q also move. In doing so, the other components of the applet change in accordance with these manipulations. The power here is that students can interact with a wide variety of examples and see if the conjectures they make hold up to empirical investigation. In addition, the student can move the corners of the box, in the lower right-hand corner of the applet, and see if the box will actually close or not, an important aspect if you want the box to hold something. The last elements in this applet are the two different graphical indicators of maximal volume. The one graph shows the volume with respect to P or Q while the other is held constant and the bar graph next to it displays the percentage of maximal volume obtained by the current configuration. If the volume is too large, one can resize the vertical unit, a yellow point denoted by U, to get the graphs comfortably into the grey viewing window.

As students play with these to attempt to improve their maximal value score, they will be led to ask a variety of questions, such as:

• Is there a relationship between the cut length and the position of the cut that maximizes the volume?
• Under what conditions does the volume drop to zero?
• Is there a best way to orient P and Q so that the maximum is achieved?
• How do the two graphics on the bottom left-hand side interact with each other?

Figure 8: The first Box Problem applet

Warning: The first Box Problem applet page, entitled ClosedBox, is best viewed in 1024 x 768 resolution or greater and may take up to a minute to load.

The goal of this exploration is to help students recognize that even though this problem may at first appear to be a problem involving two variables, the cut length and the position of the cut, focusing on maximizing the volume allows one to turn the problem into a problem of one variable. By exploring the applet, students should find that if the cut length is held constant and the position of the cut is allowed to vary, then the volume of the corresponding box is described by a constrained quadratic equation (expressed by the blue graph), and if the position of the cut is held constant while the length of the cut is allowed to vary, the volume of the box is described by a constrained linear equation (expressed by the red graph). The constraints result from the physical constraints on the box as well as whether, when the box is constructed, it will hold its contents, i.e. the flaps meet or overlap so there are no holes in the box. All it takes is some imagination after working through multiple examples and noticing the relationship between these two variables that maximizes the volume for any particular condition. From our perspective, being able to explore, quantify, and utilize is an important aspect of learning mathematics.

#### The first Box Problem activity

This applet was designed to help students come to a realization that for any positioning of the fold, the maximum volume can be best achieved when the cut length is set to one-half the smallest width of the box. Not only can students disassemble the box by pulling on the corner points, the applet helps students investigate how constraints affect the box problem. For instance, if the cut length exceeds the minimum of the length and width of the box, then the box will not close because the flaps created by the cuts will interfere with each other. Alternately, if the cut length is less than half of the minimum of the length and width of the box, then the box will not close because the flaps created will leave a hole in the box, thereby allowing the contents to fall out.

So, how does one accomplish this? In the applet, the points denoted with a yellow dot are those that are moveable. Perhaps, the first question that students should be asked concerns the midpoint fold. In particular, why does one of the folds need to occur at the midpoint? Another question should focus on the symmetry. In fact, asking ''what happens if the two folds aren't symmetric about the midpoint fold?'' can yield interesting conversations amongst the students. Getting back to the movable points, a student can set a position for where the cut is to occur and then vary the length of the cut. We suggest students carefully record the data from their various trials where the cut position, $$m( \overline{AP})$$, is set and the cut length, $$m( \overline{BQ})$$, is changed to maximize the volume for that particular cut position. The last row of each Trial is for the student to search for the maximal cut length and the corresponding cut position. The final trial is for an open exploration of a piece of cardboard of the student's choosing.

$$m( \overline {BQ})$$ $$m( \overline {AP})$$ $$m( \overline {PM})$$ $$m( \overline {QQ'})$$ $$m( \overline {BB'})$$ $$m( \overline {AA'})$$ % of max volume 1.5 3.0 4.0 5.5 8.5 14.0 97.75% 3.5 3.5 8.5 14.0 4.5 2.5 8.5 14.0 8.5 14.0 2.5 2.5 10.0 10.0 1.5 3.5 10.0 10.0 4.25 0.75 10.0 10.0 10.0 10.0 3.0 2.5 8.5 11.0 2.75 2.75 8.5 11.0 4.0 1.5 8.5 11.0 8.5 11.0

Exploring various positions of a cut, should lead the students' to a conjecture that for any position of a cut, the maximum volume of the corresponding box is obtained when the cut length is half the length of the shorter length between $$m( \overline {AP})$$ and $$m( \overline {PM})$$.

There are so many different questions that can be asked when students are interacting with the first Box Problem applet. In addition to those already mentioned, one can ask:

• Under what conditions is the blue graph connected above the x-axis?
• Under those conditions and for a particular length and width, what are the dimensions of the box with maximum volume?
• Why does the red graph always appear as a slanted portion? What mathematical meaning does this red graph have? Does this slanted portion ever change slope? If so, why and if not, why not?

It should be mentioned that we have found that if students have experienced the open box problem, there is a tendency of students to either focus on the length of the cut or the position of the cut but not typically both. Some students want to make the controlling variable the position of the cut and the resulting calculus computations are more difficult but not impossible. However, if we were to focus on the computations and not a deep exploration of the problem, students would miss a perfect opportunity to look beyond the numbers and see relationships that are intrinsically interwoven in this problem.

### The Second Box Problem

#### Introduction

After exploring the first Box Problem applet, some essential elements of moving a seemingly two-variable problem into a single variable problem have come to light. In particular, students are now prepared to explore the single-variable box problem. Specifically, the introduction provided by the first box problem applet has provided a means of reducing the complexity of the problem. Instead of having two variables with which to contend, we have found that without loss of generality, we can set the position of the cut to be twice the length of the cut.

What exactly does this provide? We can now express the volume of the box with respect to a single variable thereby opening this problem to the techniques from Calculus 1. In particular, we can turn our attention to maximizing the volume of a box based upon the cut length. Now, the second Box Problem applet, will focus its attention on looking at

$V(l) = \left( {B - 2l} \right)\left( {{1 \over 2}A - 2l} \right)\left( {2l} \right)$

where A and B are the correspondent length and width of the rectangular piece of cardboard and l corresponds to the length of the cut.

We should note that this description is equivalent to the formula presented in Dundas (1984) of

$V(l) = T\left( {{1 \over 2} - T} \right)\left( {{A \over l} - T} \right)$

• $$T = 2l$$
• $$l_{Dundas} = A$$
• $$B = w = {{A_{Dundas} } \over {l_{Dundas} }}$$

In addition, the l of Dundas (1984) needs to correspond to the position of the cut instead of the length of the cut.

#### The second Box Problem applet

At first glance, this applet, ClosedBox2, contains many of the same components as the first Box Problem applet; however, the cut length determines the positioning of the cut so that in each case the box volume is relatively maximized.

Figure 9: The second Box Problem applet

Warning: The second Box Problem applet page, entitled ClosedBox2, is best viewed in 1024 x 768 resolution or greater and may take up to a minute to load.

In this applet there are a variety of elements that can be seen. First, the point P is no longer adjustable but is rather determined by the length of the cut defined by the segment BQ. In addition, the grey box in the lower left-hand of the applet contains a dynamic graphical depiction of the functional relationship between cut length and volume. That is, it contains a graphical depiction of

$V(l) = \left( {B - 2l} \right)\left( {{1 \over 2}A - 2l} \right)\left( {2l} \right)$

where l corresponds to $$m( \overline {BQ} )$$ and currently B = 8.5 and A = 14.0.

One element that students need to grapple with when working with this particular applet is the graphical depiction of the function. In particular, graphing $$V(l)$$ using a graphing calculator or computer algebra system yields a figure similar to the following:

Figure 10: Graph of the Box Problem Function

The graphical depiction of the function presented in the applet is a truncated version of the one in figure 10. Students will need to come to grips with the fact that the applet is only concerned with the volume of boxes that are physically constructible whereas the graph of the function shown in figure 10, does not necessarily concern itself with the constructability of the box. Instead, it provides a graph of the functional relationship between an independent variable l and a dependent variable V. In essence, this graph in figure 10 is less concerned with cut length and volume and more concerned with expressing the relationship for all possible values of l, irrespective if these values are possible cut lengths or if those cut lengths yield appropriate volumes.

Too often, we see students focused on the algebraic elements of a problem without considering the physical (or mathematical) constraints on that problem. We seek in this applet to guide students to grapple with the interplay of these two seemingly disparate forces. In turn, we hope to lead them to reconcile for themselves how functional relationships that model real-world phenomena require a careful examination of the domain for which that relationship actually does model the phenomena. For instance, students might at first think that l's only restriction is that it must be less than $${1 \over 2}m( {\overline {BB'} } )$$ since a cut cannot exceed half the width of the cardboard and maintain its connectedness. However, there is another constraint: the cut length, l, cannot exceed $${1 \over 4}m( {\overline {AA'} } )$$ . This ''hidden'' constraint comes directly from the relationship of cut length and position of the cut and depends on the relationship between length and width of the rectangular piece of cardboard.

#### The second Box Problem activity

This applet was designed to help investigate the interrelationship between cut length and volume. In particular, the previous interactions with the first Box Problem applet allowed the student to recognize the relationship between cut length and position of the cut. Using this, the student can build a functional relationship, based on a single variable, relating cut length and volume. This applet is designed to explore that functional relationship and help students come to a realization that in modeling real-world phenomena, a careful examination of the domain of a the functional relationship is necessary.

How can one ask questions to help students interact with the second Box Problem applet in meaningful ways? From our experience, the first question that probably should be asked involves the relationship between the first Box Problem applet and this second Box Problem applet. In particular, we can see in this new applet that the point P varies as the point Q moves, what information drawn from the first Box Problem applet allows us to control the point P? Another question should focus on constructing the volume of the box based upon the length of cut. Essentially, a series of questions to help lead students to developing a functional relationship between volume and cut length, such as:

• How does one determine the volume of a rectangular box?
• What would be a description for the length of the box?
• What would be a description for the width of the box?
• What would be a description for the height of the box?
• How can you use these descriptions to build a functional relationship between volume and cut length?

Inherent to these questions are questions about what are the constants and what are the variables? Such a question is important when students are faced with multiple letter-based descriptions such as those expressed in the formula for $$V(l)$$ :

$V(l) = \left( {B - 2l} \right)\left( {{1 \over 2}A - 2l} \right)\left( {2l} \right)$

Once students have developed appropriate functional relationships between cut length and volume, a variety of questions relating to that function can be asked. In general, we typically ask students to investigate the graph of the function on a hand-held graphing calculator. And then, we start asking various questions to help the students compare and contrast the graphical representation produced by the graphing calculator and that of the applet. For instance, we might ask:

• Why does the graphing calculator seem to show you more of the graph than the applet does?
• Why does the applet truncate the graph?
• What conditions should be on the domain of the function and how do they relate to physically constructing a box?

So far, these questions have focused primarily on a static rectangular sheet of cardboard. That is, we have not asked questions that forced students to think about varying the size of the cardboard.

• For different lengths or widths, the applet's graph seems to change shape near the right-hand terminus, what mathematical reason can you provide for this change or provide an argument that it does, in fact, not change?
• Are there two (or more) non-isomorphic sheets of cardboard, so the maximal volume is the same? If so, identify them and if not, explain why not.
• Are there two (or more) non-isomorphic sheets of cardboard, so the placement of the maximal cut is the same? If so, identify them and if not, explain why not.

Here, the questions are designed to force students to think beyond the original question and attempt to distill commonality and explore generalizations. Stretching students to think beyond and to ask ''what if'' questions is an important aspect of interacting with this applet. We sought to design the applet to support both the investigation of a specific scenario as well as a broad range of extended questions related to the original problem.

### Reflections and Extensions

This paper has attempted to illustrate how applets can broaden the investigation of mathematical relationships and reach beyond mere algebraic manipulations. The ability to investigate, conjecture, test, and pose new questions in an electronic environment is important to the development of understanding. Beyond this, such investigations are enjoyable for students. For instance, students mentioned the box problem in their weekly Calculus 1 journals and two examples are provided below:

''This last week we learned of Indeterminate terms and L' Hopital's rule. Did quite a few examples on how to perform the rule, contionued [sic] this for four days, then I think we made boxes. I believe that I learned how to do L' Hop's rule fairly well, hopefully I could do well on a test, and the box making was fun and surprised me in that you fit it into the lesson. I enjoyed the interactivity.'' - Charles
''This week we learned about maximum volume of boxes and solved a pretty cool problem involving James Bond and Bombs. I love these sorts of problem-solving and I hope that we do more of it in the future.'' - Dan

Our goal has always been to get students to think that problem-solving is fun and the interaction between building physical boxes and the computer applets helps students draw essential mathematical connections. The lessons that can be drawn from these activities bolster students' perceptions that mathematics is eminently useful and applicable to real-world phenomena much more than the typical box problem shown in many of the calculus texts available on the market.

But is that as far as one can go into the world of boxes? We should also mention that this problem of optimal box building could be further extended by including the tab to affix the sides of the box to each other without the use of a taped joint. At one level, our discussion has focused its attention on construction of a minimal shipping container of the Regular Slotted Container type; however, if we were to introduce the requirement of a tab it does impact the problem. It should be noted that there are general governmental guidelines for the width of such a tab but 1 1/4 - 1 1/2 inches is typical.  For specific guidelines about tab regulations, one should check the  Fiber Box Handbook (Fiber Box Association, 2005), the National Motor  Freight Classification (NMFC) Item 222 (American Trucking  Association, 1970) for common carrier motor freight shipment, or the  Uniform Freight Classification (UFC) Rule 41 (Dolan, 1991) for rail  shipment.  In any case, we think that after exploring the box problem through these applets, students might be better prepared to think how to construct a RSC containing tabs (as shown in figure 5) with maximum volume?

Beyond even looking at a tabbed RSC, the world of boxes is much more encompassing. In this paper, we have focused our attention on the most popular box, a regular slotted container, but there are a host of other types of boxes fabricated from a single piece of corrugated cardboard that students can investigate, including:

• A Full Overlap Container (FOL), see figure 10a, that is resistant to rough handling since all flaps are of the same length (the width of the box) and when closed, the outer flaps come within one inch of complete overlap,
• A Five Panel Folder (FPF), see figure 10b, that features a fifth panel used as a closing flap completely covering a side panel and includes end pieces composed of multiple layers that provide stacking strength and protection of long articles with small diameter, or
• A Once Piece Folder (OPF), see figure 10c, typically used books and printed materials since it has a flat bottom with flaps forming the sides and ends along with extended flaps that meet toform the top (GoPackaging.com, 2006).
 (a) (b) (c)

Figure 11: Other types of boxes manufactured from a single piece of corrugated cardboard

Each of these boxes would provide an opportunity to extend investigation and gain a better sense of the issues behind box construction. We certainly hope that this paper hasn't boxed you in but rather opened your eyes to the ways boxes can be used to help students see the ability of calculus to play a meaningful role in examining real-world problems.

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