The Box Factory
On this web page you will investigate the different
ways to construct an open-top box from a sheet of paper. This
exploration will reinforce the concepts in Lesson 7.6 of Discovering
Advanced Algebra: An Investigative Approach.
This sketch shows a green sheet of paper from which you
can make an open-top box by cutting out congruent squares from each
corner. You can drag points W, L, and x
to adjust the width, W ,of the paper, the length, L,
the paper, and the edge length, x, of the square. You'll also
see a picture of the box formed from the paper.
- Adjust the width and length of the paper so that it
is 16 units by 20 units. Make a table with several different values of x
and the corresponding volumes.
- Can you write functions that give the width of the
box and the length of the box in terms of x?
- Write a function that gives the volume of the box in
terms of x.
- For what value of x does the box have the
largest volume? The smallest volume?
This sketch is similar to the first sketch, but now it
also includes a graph that shows the volume of the box plotted as a
function of the length, x. As you change W and L,
the graph will change. The graph shows the volume of the box as a
function of the box height x. When you change x,
you will see the point on the graph move.
- What do you think is the degree of the function shown
in the graph? Give some reasons to support your answer.
- How can you find the box with the largest volume for
a given height and width? Explain.
- Change W and L. How do these
values affect the graph of the volume function?
- For H
= 16 and L
= 20, write an expression that gives the volume as a function of the
length, x. Can you factor this polynomial expression? What
the three roots of this polynomial? How can you interpret them in terms
of this problem? What domain of x
makes sense for modeling this situation?