# "Inverse" Box Problems

Objective: To provide practice with word problems related to geometric figures. The sequence of "box problems" was designed to give students practice interpreting the given geometric characteristics of a box in order to determine the smallest rectangular piece of cardboard from which the box could be made.

Level: Precalculus or calculus.

Prerequisites: Students need to know how to compute the area of a rectangle and the volume of a rectangular solid. The Pythagorean Theorem is needed for the pyramid box.

Platform: Browser and QuickTime player.

Instructor's Notes: In a calculus course the application of derivatives to optimization problems requires that students have experience in setting up equations that describe a variety of physical situations. Some students have difficulty creating a visual image from a verbal description and this makes the problem of developing equations to algebraically describe the situation a challenge. Experience in developing algebraic models for physical situations prior to working with optimization problems, which require algebra, geometry, and differentiation techniques, can provide a foundation to better equip students for this important aspect of problem solving.

Here we present a sequence of geometric problems in which the object to be modeled is a box to be constructed in a particular way from a rectangular piece of material.  Since the dimensions of the box are specified we are not solving an optimization problem involving volume, rather we want to develop an algebraic model that can be used to determine the dimensions of smallest rectangular piece of cardboard that can be used to construct the particular box. Only algebra is required. Students must interpret geometric information given for the box in order to appropriately assign values to portions of the rectangle that circumscribes the unfolded box. It is possible to determine the length and width of the smallest rectangular piece of material without writing equations, however, a goal is to have students develop equations since that is the skill needed in calculus.

We call these modeling exercises "Inverse Box Problems". We present seven types of boxes. The corresponding modeling problem (approximately) increases in difficulty.

Type 1. A box with no top and volume 1,428 cubic inches is shown in Figure 1. This box is formed from a rectangular piece of cardboard by cutting the same size square from each corner. Figure 2 displays a diagram of the cardboard with squares removed from the corners which when folded along the dashed lines will form the box.

 Figure 1. Figure 2.

Determine the length L and width W of the smallest rectangular piece of cardboard that can be used to construct the box shown in Figure 1.

Type 2. A box with a top and volume 12,000 cubic inches is shown in Figure 3. The dashed line in the center of the top indicates that the top is formed by two pieces coming together as depicted in the diagram in Figure 4. As we continue to "unfold" the box in Figure 4 we obtain the diagram in Figure 5 in which the dashed lines indicate the folds that are made to form the box.

 Figure 3. Figure 4.
 Figure 5.

Determine the length L and width W of the smallest rectangular piece of cardboard that can be used to construct the box shown in Figure 3.

Type 3. A box with a top and volume 528 cubic inches is shown in Figure 6. The box is formed by folding
cardboard as shown in Figure 7.  As we "unfold" the box in Figure 7 we obtain the diagram in Figure 8 in
which the dashed lines indicate the folds that are made to form the box.

 Figure 6. Figure 7.
 Figure 8.

Determine the length L and width W of the smallest rectangular piece of cardboard that can be used to construct the box shown in Figure 6.

Type 4. A box with a top and volume 3,640 cubic inches is shown in Figure 9. In Figure 10 we show the unfolding of the box.  When we completely unfold the box we obtain the diagram in Figure 11 in which the dashed lines indicate the folds that are made to form the box.

 Figure 9. Figure 10.
 Figure 11.

Determine the length L and width W of the smallest rectangular piece of cardboard that can be used to construct the box shown in Figure 9.  (Hint: Figure 11 can be created by first folding the rectangular piece of cardboard in half to form a rectangle with dimensions L/2 by W and then cutting a square of side 5 inches from each corner and then unfolding the cardboard.)

Type 5. A box with a square base and volume 9,000 cubic inches is shown in Figure 12. This box has a double layered top. In Figure 13 we show the unfolding of the box. When we completely unfold the box we obtain the diagram in Figure 14 in which the dashed lines indicate the folds that are made to form the box.

 Figure 12. Figure 13.
 Figure 14.

Determine the length L of the smallest square piece of cardboard that can be used to construct the box shown in Figure 12.

Type 6. A box in the shape of a pyramid and volume 8,000/3 cubic inches is shown in Figure 15. (Recall that a pyramid has a square base and its volume is given by V = 1/3 (area of the base) (height).) Figure 16 shows the triangular sides of the pyramid unfolded giving a star-shaped figure.
 Figure 16. Figure 17.

Determine the length L of the smallest square piece of cardboard that can be used to construct the box shown in Figure 16. Hint: Inscribe Figure 17 in a square as shown in Figure 18. We can construct Figure 17 from Figure 18 by cutting away the four congruent shaded triangles. In order to determine L given the values of s and h we need to introduce another parameter, x, as shown in Figure 19 and observe that the height x of the shaded triangles is located at the midpoint of the side of length L. Next use  the right triangle with hypotenuse s in Figure 20 to obtain an expression for s in terms of L and x. Now we introduce another parameter t in Figure 21. Use the Pythagorean Theorem to express t in terms of x and L. The expression for t gives the length of the edge of the pyramid from the top to one corner of the base. In Figure 22 we have a right triangle with hypotenuse t, one leg is h and the other leg is the distance from the center of the base to a corner of the base. (See Figure 21 for another view of the "other" leg.) Using the numerical values for s and h, after some simplification you should have two expressions that contain just L and x; now use algebra to determine both x and L.

 Figure 18.                        Figure 19.                     Figure 20.                    Figure 21.
 Figure 22.

Type 7. For each of the boxes Type 1 through Type 6 tape or some type of adhesive is required to hold the shape of the final configuration. Here we investigate construction of a type of box that holds itself together; that is, it is self-standing. To illustrate the construction of such a box we provide a QuickTime movie. In this movie you start with a sheet of cardboard 26 by 15 and perform some cutting and folding. Using the movie, which can be paused, reversed, and moved forward you are asked to determine the volume of the resulting box. So keep some notes by answering the questions that are included so that at the end you can compute the volume of the box. To start the movie click on the box in Figure 23.

 Figure 23.

Discussions and Extensions:

Auxiliary Resources: A toolbox of visual aids for geometrically oriented word problems is available at
http://mathdemos.org/mathdemos/wordproblemeqn/wordproblemeqn.html  The visual tools of this demo are designed to help students develop equations that provide an algebraic model for a variety of problems. A discussion and an outline of steps to use in geometric word problems is provided. Examples and a gallery of animations are included.

Credits:  This demo was developed by

David R. Hill
Mathematics Department
Temple University

and is included in Demos with Positive Impact with his permission.

DRH 5/25/2007     Last updated 6/5/2007

Visitors since 5/30/2007