An examination question that did not square up
You have learned how to minimise something by making its derivative
vanish, and spent hours practising the calculus techniques involved.
But when mathematics is applied to real world problems,
it is not only the techniques that must be correct.
You have to make sure you are
working on the right problems in the first place.
The problem with real life is that things are never quite as simple as one might wish. I realised this recently when one of my favourite classes started to give me a hard time over my solution to an old examination question. It was of the sort of question for which differentiation, and its ability to determine the minimums, maximums and points of inflection of a curve, was ideally suited to solve.
It is required to make an open box with a square base and rectangular sides from the net shown in Figure 1. If x is the length of a side of the square base, and the box is to contain a given volume V, express the area of the net in terms of V and x. Also, what ratio of box height, h, to x will use the minimum area of net ?
At this point, you may like to have a go at solving the problem for yourself.
The volume of the box is
Which is rearranged as
The area of the net is
Substituting (2) into (3) gives the answer to the first half of the question as
As V is specified from the outset, it is a constant and we can thus differentiate (4) with respect to x to get
Substituting (1) into (5),
For the required minimum, the derivative, (6), must equal zero and so the final answer is that h : x = 1 : 2.
The Reality Gap
I was, like I imagine most people who answered that question in an examination were, happy with the solution. However, the lesson after I had demonstrated it on the whiteboard, one of my class presented me with a 5OO gram box of 'Sainsbury's Biscuits for Cheese' !
First, my attention was drawn to the fact that this box had a square base. Then it was opened (the biscuits, alas, had gone) to reveal that this box was, in fact, made from two open topped boxes of the type considered in our previous lesson. See Figure 2.
The lower half had a square base of slightly shorter side length than the top, such that the two fitted snuggly together. Out came a ruler. The square was of side 22cm. The 1 : 2 ratio result predicted a height of 11cm. Triumphantly, it was pointed out to me that this box was only of height 5cm. The 'Biscuits for Cheese Reality Ratio' was 1 : 4.4 and not 1 : 2. 'Maths', it was announced, 'does not square up with reality.'
Clearly, such a massive discrepancy between our mathematical solution and what is happening down at the local supermarket needs an explanation. Before I give it, however, I would like you to ponder the problem. I shall even give some help:
? ? ? Are non mathematical considerations such as design fashion to blame ?
? ? ? Is the look of the ideal box, mathematically speaking, ugly to the human eye ?
? ? ? Is the nature of the product dictating the dimensions of the box ?
? ? ? The biscuits come within two open boxes rather than one - does this explain it ?
? ? ? Is it that the sheets of card used to make the boxes only come in standard sizes ?
? ? ? Is it to do with building the box in reality ? Surely we need some "tabs".
Now, either make your choice from the above or give a reason of your own.
Mathematics Review, November 1992, Volume 3, Number 2.
© Philip Allan Publishers Limited. ISSN 0957-1280
All figures redrawn, © Martin Hansen, December 2007
New photography, © Martin Hansen, December 2007
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