Objective:
To provide practice with optimization problems related to geometric figures.
A sequence of "box problems" is designed to help students develop an
equation for the volume of a box constructed from a rectangular sheet of
cardboard by cutting away portions and folding the remaining portions to
construct a box in a particular way. The equation for the volume is to be
formulated in terms of the dimensions L and W of the piece of cardboard and
a parameter x, often related to the height of box, so that calculus can be
used to determine the value of the parameter that will maximize the volume.Level:
Calculus students in high school or college.
Prerequisites:
Differentiation and concepts related to optimization.
Platform: PC or
Mac; Excel demos for each type of box are included.
Instructor's
Notes:
In a calculus course the application of derivatives to
optimization problems requires that students draw on a number of skills in
order to construct an appropriate equation to optimize.
For some students the physical situation itself is a challenge to visualize
properly in an effort to construct corresponding equations. Here we present a
sequence of physically oriented problems in which the geometry is a familiar
box which can be constructed from a rectangular piece of cardboard by
cutting away portions and folding remaining portions. By varying the type of
box to be constructed it is necessary to develop a variety of relationships
between the size and type of portions cut away and the dimensions L by W of
the piece of cardboard.
Each demo has an Excel file that shows a
picture of the type of box to be constructed and a labeled diagram of the
unfolded box with information about the sections that are folded in the
construction process. The user can enter values for the length and width of
the piece of cardboard. The interactive part of the Excel file uses a slider
to generate a graph of the volume of the box that can be constructed from the
piece of cardboard for various choices of a parameter x related to the
height of the box. The graph can be used to estimate the value
of the parameter that will give the largest volume for the box.
Students are to asked to use the figures to
construct a formula for the volume of the box in terms of L, W, and x
(the parameter), then to use calculus to find the exact value of x so
that the box has maximal volume. (As a check they can compare the estimate
from the generated graph with their result.) Finally they are asked to
determine the corresponding maximal volume.
Since the dimensions L and W of cardboard can be
changed a variety of different size boxes can be generated.
If students need practice in developing ideas associated
with optimization problems there are several auxiliary resources available in
the Demos with Positive Impact collection.
1. The demo
Constructing Equations from Word Problems provides a toolbox of visual
aids for geometrically oriented word problems. There is an accompanying
gallery of problems that contains animations and some Java Applets. Click
here to go to this resource.
2. The demo Inverse Box Problems
presents a sequence of physically oriented
problems in which the geometry is a familiar box and a particular rectangle
related to the unfolded box. Since the dimensions of the box are specified
we are not optimizing 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. Here 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.
Click
here to go to this resource.
A gallery of Box Maxmin Problems follows. Just click on
the box figure to execute or download the Excel file that accompanies the type
of box displayed.
Gallery of Box Maxmin
Problems
|
 |
The box has no top. |
 |
The top flaps are joined to form the top of the box. |
|
 |
The top is in one piece. |
|
 |
This box is like a suitcase. |
|
 |
This box has a double layered top. |
|
 |
This box has a pyramid shape. |
|
 |
This box is self-standing; that is, no tape is
needed to keep the shape of the box. |
