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Vandaag is het 8 December 2022

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Milk bottles - math puzzle


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Suppose you have two bottles with equal perimeter, thickness and height. One bottle is cilindrically shaped, the other one has the shape of a rectangular box with a square shaped bottom. In which of the bottles fits more milk?


The volume (V) of a cilindrically shaped bottle is equal to the area of the bottom (A) times the height (h). So I = A*h.

A circle with perimeter C has a larger area than a square with perimeter C. Therefore, in the cilindrically shaped bottle fits more milk. We will proof this below.

The length of one side (l) of the square is the perimeter (C) divided by 4. So l = C/4. Hence the area of the square is A = C*C/16. The volume of the rectangular box is hence h*C*C/16.

The radius (r) of the bottom of the cilindrically shaped bottle is C / (2 * pi). The area of the bottom (pi * r * r) is hence C*C / (4*pi). So the volume of the cilindrically shaped bottle is h*C*C / (4*pi).

Because 1/(4*pi) is larger than 1/16 (pi is about 3,14) the cilindrically shaped bottle has the largest volume.
One can even proof that given a circumference C, a circle is the shape with perimeter C that has the largest possible area.