Design a Better Drink Can

Stephen ARNOLD
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Adapted from an algebraic modelling activity on the HSC Online website.

A standard can of drink holds 375 ml.
Most are around 6cm across the base, and 13 cm high.

But is this the BEST size?

Since the cost of materials is most closely related to the surface area of the can, we would like dimensions that give the greatest volume for the smallest surface area.

Drag the point shown to change the radius and observe the graph to see how the surface area (divided by 100!) changes! Try different volumes.

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Try and find the algebraic function which matches this situation.

What assumptions have we made in building this model, and how might these be minimised?
Would a volume of, say, 400 mls make more sense - allowing for some air in the top of the can?
The top and bottom of the can are thicker than the sides - perhaps our surface area formula shold be adjusted to allow for, say, double thickness at top and bottom?

Steve Arnold, 15/10/2006, Created with GeoGebra

This problem may also be modelled using CabriJr on the TI-83/84 series calculator. Click on the graphic below to download the file and try it yourself!

Try having students work in pairs: one generating data points using the CabriJr model and the partner entering these into lists to plot and then to help build their algebraic model!

And just wait until you try this with TI-nspire CAS!

Extension...

I decided to extend this problem a little further, and found that I had five different can sizes in my pantry! (Note that my data opposite includes a can of soft drink which is not shown).

Can you pick the odd one out?

Are any of these cans optimal?