A cylindrical can with volume V can be made in different ways. The cylinder can be wide and short (large radius and small height), or it can be narrower and taller (small radius and larger height). Which shape uses the least material? That is, which can has the least surface area? Solve for the ratio of the height to the radius. This video presents the optimal ratio.
My blog post for this video
I first learned about this problem from "Aha! Solutions" by Martin Erickson: http://www.maa.org/press/books/aha-solutions
I also came across this problem from the DataGenetics blog, which also did an analysis of actual cans. Many commercial products are actually pretty close to the optimal size. The soup can is nearly a perfect design!
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