The Department of Mathematics and Science Education

Allyson Hallman

Bank Thief: How big is my box?

I have stolen gold coins from the National Bank of Georgia. I will soon be rich if only I can avoid the police. Unfortunately, they are on to me so IÕve got to hide the loot. But I have little time and resources. If you can help me, I will cut you in for 25%.

The Problem:

I intend to bury my gold coins. But I if I just dump them in a hole, I may not be able to recover them all. I live by a lake with ducks and if I donÕt protect my coins in some type of enclosure (a.k.a. box) I may find myself digging through mounds of duck poo to recover them. I must construct a box. The only tools I have available are two 8 in. x 5 in. pieces of sheet metal, wire cutters that will go through the sheet metal, a 12 in. ruler, and of course welding gear (doesnÕt everyone?). I need to cut equal sized squares from the corners of one piece of the sheet metal and then by folding up the resulting flaps and welding them together construct an open box.  The second piece of sheet metal will form the top of our box.

Having trouble picturing this? Click here to explore the folding box in GSP.

Consider this picture of our sheet metal:

What do we know about the size of squares we cut out? Ah ha. The squares cannot exceed the lengths of our box, so the square must be smaller than 5. In fact it must be less than 2.5 in. (more than 2.5 in. and the lengths of the two squares cut from the 5 in. side would exceed 5 inches.)

And with the squares cut out.

LetÕs consider the formula for the volume of the box.

height = length of a side of a square

width = 5 – 2(height)

length = 8 – 2(height)

LetÕs consider what is going with an excel spread sheet:

 Original Sheet Metal Dimensions Length Width 8 5 x =5-2x =8-2x V = x(5-2x)(8-2x) Height Width Length Volume Is 18 the maximum volume? 1 3 6 18 2 1 4 8 3 -1 2 -6 Well, of course this is negative; we've exceeded  our maximum height allowance. Ok, let's try with a smaller incremental changes in our height. Height Width Length Volume 0.1 4.8 7.8 3.744 0.2 4.6 7.6 6.992 0.3 4.4 7.4 9.768 0.4 4.2 7.2 12.096 0.5 4 7 14 0.6 3.8 6.8 15.504 0.7 3.6 6.6 16.632 0.8 3.4 6.4 17.408 0.9 3.2 6.2 17.856 1 3 6 18 1.1 2.8 5.8 17.864 Click above to explore further in excel. 1.2 2.6 5.6 17.472 1.3 2.4 5.4 16.848 1.4 2.2 5.2 16.016 1.5 2 5 15 1.6 1.8 4.8 13.824 1.7 1.6 4.6 12.512 1.8 1.4 4.4 11.088 1.9 1.2 4.2 9.576 2 1 4 8 2.1 0.8 3.8 6.384 2.2 0.6 3.6 4.752 2.3 0.4 3.4 3.128 2.4 0.2 3.2 1.536

Can we confirm this with a graph?

Here is a scatter plot from excel:

And graphing the equation

Volume = x(5 – 2x)(8 – 2x)

We have the following:

So if it easy to see that the maximum of the graph on domain (0, 2.5) is in fact 18, and that this maximum occurs at 1.

Thus the maximum volume we can achieve with our box is 18 cubic inches and the dimensions of that box are:

1 inch x 3 inches x 6 inches, which you can construct by cutting 1 inch x 1 inch squares from the corners of the original piece of sheet metal.

Click on the graph above to explore in graphing calculator.

Want to have more fun??

What about boxes constructed from different sizes of sheet metal?

In the GSP file you can modify the dimensions of the original sheet metal and explore. Have a ball.