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Tin Tight

What's the most efficient proportion for a 1 litre tin of paint?

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Paint Rollers for Frieze Patterns.

Proofs that there are only seven frieze patterns involve complicated group theory. The symmetries of a cylinder provide an easier approach.

Tubular Stand

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

The corners of the stand have to be considered and this means that each individual section is not a cylinder. However a little careful manipulation will reveal that the problem is far from insurmountable.

For the generalisation it might help to reduce the number of variables you work with to two at a time...

What is the relationship between the length and the volume if the radius is fixed?

Is there a way you can represent the relationship without just trying to describe it in words?