Maximizing Volume of a Cereal Box
Date: 07/08/99 at 23:10:43 From: Phil Baker Subject: Maximizing volume of a box I have a follow-on question to a standard max/min problem. The problem is the one where one has to find the maximum volume of a box, given a certain-sized piece of cardboard (i.e., how big a square should be cut from each corner before folding the cardboard into a box?) I can show the calculus for how to find the maximum volume. My question is, does anybody have any insights about why (for instance) cereal boxes are the size they are? Is it due exclusively to maximizing volume, or are there other considerations - manufacturing, marketing, ergonomics, etc? Thanks for any insight you can provide.
Date: 07/09/99 at 17:14:42 From: Doctor Peterson Subject: Re: Maximizing volume of a box Hi, Phil. I have no professional knowledge of this, but I can imagine lots of things besides maximizing the volume that would go into this decision: Manufacturing: try to maximize volume, within various constraints size of blank cardboard available - make box fit size of cutting and printing machines size of shipping cartons and pallets ease of making bag inserts to fit (harder if square bottomed) Marketing: try to maximize AREA for printing logos, ads, and required text tradition - don't make boxes too different from old boxes Ergonomics: can't be thicker than a small person's hand, so it can be held box should be reasonably sturdy when open Etc: realities of manufacturing: overlaps and waste modify the constraints I tried some sample constraints to see what they would do. If we just try to minimize material for a given volume, with the assumption that the entire top and bottom and one side overlap, so that the area of material is A = 2hw + 3hd + 4wd then the optimum ratio is h:w:d = 4:3:2, not surprisingly, since that makes the three terms equal. That's not too far from the typical box - at least better than the optimum for a closed box without overlaps, which is a cube. Then I tried including waste for a box cut this way: +---+-----------+---+-----------+---+ d|xxx| |xxx|:::::::::::|xxx| +---+-----------+---+-----------+---+ | | | | |:::| | | | | |:::| | | | | |:::| h| | | | |:::| | | | | |:::| | | | | |:::| | | | | |:::| | | | | |:::| +---+-----------+---+-----------+---+ d|xxx| |xxx|:::::::::::|xxx| +---+-----------+---+-----------+---+ d w d w d (xxx = discarded parts or flaps, ::: = overlapped ends for gluing) so that the area to be minimized is A = (h + 2d)(2w + 3d) = 2hw + 3hd + 4wd + 6d^2 This gives h:w:d = 4:3:1, if I did it right, which is even closer to what a cereal box is; this makes sense, since minimizing waste would make us want d to be smaller than in the other case. This almost entirely explains the shape, which surprises me. But I suspect tradition is the main influence on box shape today, unless government regulations play a role too. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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