The Solution
None of the given suggestions explains the totally wrong 1 : 2 ratio. If you picked one of them then I do sympathise. They are tempting because they could explain a small deviation away from a good mathematical answer.
The problem is, however, that the 1 : 2 mathematical answer is based upon a hopelessly naive understanding of how such boxes are made. If you worried over the issue of "tabs" then you had taken a first step in the right direction, but the 'tabs' could be made small and so only alter the result a little. On the real box, however, they are huge !
Look at Figure 3.
It is the net of our box modified so that the 'tabs' are included as the red corner squares.
Figure 3
Now look at Figure 4.
It shows how many such nets would be cut from a large sheet of card.
Figure 4
If we cut out copies of our ideal Figure 1 net, those red parts would be lost. We do need 'tabs' and they may as well be huge as they are being made from material that otherwise would be thrown away. Figure 1 did not, indeed, 'square up with reality' !
Now to tackle the mathematics of our new, closer to reality box. Once again, the volume of the box will be
And again its rearrangement is
The area of the net now includes an extra term:
So the first answer is
Differentiating (4b) gives
Substituting (1) into (5b),
This we set equal to zero. Multiplying throughout by x, and dividing throughout by 2, results in the quadratic
which has two factors:
Only the first of these is meaningful, and so the area of our packaging net this time has an h : x ratio of 1 : 4. This is vastly closer to the value observed of 1 : 4.4 and now, perhaps, the small discrepancy is explanable in terms of the arguments advanced earlier.
Mathematics Review, November 1992, Volume 3, Number 2.
(c) Philip Allan Publishers Limited. ISSN 0957-1280
All figures redrawn, (c) MHH, December 2007
New photography, (c) MHH, December 2007
Martin H Hansen, Shrewsbury School, Shrewsbury. SY3 7BA
All images, text and file downloads (c) 2006, 2007 & 2008, The MathMagical Software Company.