Calculus students who have solved some optimisation problems will know that a square is the optimal rectangular shape in that it maximises the area for a fixed perimeter or minimises the perimeter for a fixed area.

A slightly more difficult problem is to minimise the surface area of a cylinder with a given volume, thus minimising the cost of material to manufacture the cylinder. The question usually asks to find the required radius or height and the minimum surface area. I always ask my students to compare the height with the diameter – they are equal. This corresponds to the optimal rectangle having equal sides.

I then ask them to think about cylindrical cans of food or drink and the relationship between height and diameter. Three cans I have on hand have the following measurements:

Can |
Diameter |
Height |
Diameter/Height |

1 |
10 |
6 |
1.67 |

2 |
7 |
10.5 |
0.67 |

3 |
10 |
11 |
0.91 |

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The first two have approximately opposite relationships between diameter and height (the first has the diameter more than 1.5 times the height and the second has the height 1.5 times the diameter) while the last is close to optimal.

Why would these ratios differ so much? Besides the cost of manufacturing the can (depending upon the material that it is made from, the deviation from optimal may not be much more expensive), there may be practical considerations, for example, manufacturing constraints, optimal use of volume for transport and storage, ease of handling, available space on the supermarket shelf and consistency among product lines.

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