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.
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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.
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