Today Puzzle #759
Puzzle No. 759 鈥� Thursday 11 June 2020
A tin of beans has a cylindrical body and flat circular ends. Manufacturers like to use as little packaging as possible in the interests of the environment. The metal on the ends is about the same thickness as the metal on sides. If the shape of a tin is chosen to minimise the use of metal what should the height of the tin be compared with its diameter?
Today’s #PuzzleForToday has been set by Hugh Hunt, Reader in Engineering Dynamics and Vibration at Trinity College, Cambridge
Click here for the answer
height = diameter
We want to minimise the surface area of the tin for a given volume length of tin is L, radius of tin is R Volume V=πR²L so for a given V and R then L=V/(πR²) Surface area S=2πR² +2πRL = 2πR² + 2V/R we want to minimise this, so differentiate πR² + V/R and equate to zero (or plot a graph) so 2πR -V/R² = 0 and use V=πR²L which gives 2R = L , or L=D which is the answer.
In fact, if you measure a tin, it's closer to height = 1.5 x diameter which is because the metal on the ends is thicker. If you redo the sum for end thickness = n times side thickness you get L=nxD. This means that if the ends are 50% thicker than the sides then L=1.5D which is about right for a tin of beans.
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