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/