Mathematics in packaging design

Our household is committed to the four "R"s for purchases.  R for Reuse, R for Reduce, R for Repair, and R for Recycle.  It grieves us to put waste into the bin for non-recycleable items which go to landfill.  And we try to find alternative uses for items, rather than throw them out into the green bin.

We use old mushroom boxes, taken from the waste of our local greengrocer's shop, as boxes for storage; one box holds twelve jam-jars, and one box is a convenient size for office waste (just big enough for A4 paper and newspapers).  Latterly we noticed that the shape of the boxes had been changed very slightly.  The bottom corner of boxes used to resemble the corner of a cuboid, like the picture of the blue box below.

The newer boxes have a changed design, with the sharp corner cut away as in the black box below.  Geometrically, it is like the change from a cube to a truncated cube, with equilateral triangles in place of the box corners.

Why the change?  These black boxes will be easier to make, because the three-dimensional right-angle is a little difficult to ensure it is properly made in the plastic moulding.  Sharp corners are slightly tricky to make with the kind of plastic used for these boxes.  There are obtuse angles instead, and moulding processes will be simpler and more reliable.

So someone has spotted a problem, and devised a solution.  Where's the operational research component?  The box has to hold a specified weight of mushrooms, in this case 6 pounds.  (Not a metric size!)  So, the volume of the box cannot be reduced by very much, as the supply chain is familiar with the external dimensions of these boxes..  On the other hand, the dimensions of the cutaway corner involve some mathematics.  The "ease" or cost of manufacturing will depend on those dimensions, so perhaps somebody has applied a little modelling to determine an acceptable compromise of strength and cost, subject to the volume being within some constrained amount.


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