Making a Water Jug

Calculus Level 3

Consider a circular metal disk of radius 1 1 . Suppose all of the metal in the radial region α ≤ r ≤ 1 \alpha \leq r \leq 1 is cut off and re-shaped into cylindrical siding of radius α \alpha .

The siding is then welded to the circular base to make a metal water jug with one open end.

What value of α \alpha maximizes the volume of the water jug?


The answer is 0.57735.

Steven Chase by Steven Chase , 37, USA

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1 solution

Otto Bretscher
Dec 20, 2018

The region that is cut off has an area of π ( 1 − α 2 ) \pi(1-\alpha^2) . The jug will have a height of 1 − α 2 2 α \frac{1-\alpha^2}{2\alpha} and a volume of π α ( 1 − α 2 ) 2 \frac{\pi\alpha(1-\alpha^2)}{2} . The volume is maximal when α = 1 3 ≈ 0.57735 \alpha = \frac{1}{\sqrt{3}}\approx \boxed{0.57735}

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