The Lidless Box Problem
Using Algebra Xpresser

by

Helene Chidsey and Lou Ann Lovin

Here is the problem again for reference:
Given a rectangular sheet of cardboard 15 in. by 25 in., what size square should be cut from each corner to produce a box with a volume of 400 cu. in.? What size square would produce the maximum volume? (The square cut from each corner needs to be the same size since we want to fold each side up along the cuts to form a lidless box.)

If we let x equal the length of one side of the square, then we have the following equation for the volume of the resulting box:

.

Using Algebra Xpresser, we graphed this cubic equation to get the following picture (Figure 1).



Figure 1

Notice that the maximum volume is a bit above 500 cu. in. when x is a little below 5 in. Exactly how much above 500 and how far below 5, is difficult to see in this picture. Thus, we zoomed in to get a closer view (Figure 2).



Figure 2

From this picture, we can tell that the maximum volume is somewhere around 510 cu. in., when x is about 3.5 in.

In figure 1 the volume is 400 cu. in. when x is approximately 2 or 5 in. Again we zoomed in to get a closer view. Figure 3 is the graph on the upswing.



Figure 3

Thus when x is approximately 1.52 in. the volume is 400 cu. in.

Figure 4 is the same graph on the downswing.

On this side, when the volume is 400 cu. in., x is about 4.7 in.

Thus, we can continue to improve the accuracy by zooming in on the graph, i.e., changing the scale on the axes.


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