We are a group of math enthusiasts and we design and present our mathematical problems to societies. This week I designed this problem and I thought it might be interesting to share it with you here. If you think sharing such problems are not appropriate for this site, then I can remove it.
Here is the problem:- A spherical glass is resting on its side on a table. What is the maximum volume of water it can hold in that position?
We ignore the thickness of the glass edges.
The picture is designed and rendered in $PovRay$.
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Seyed
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Sharing this type of problems is ok for this site. Do you have any ideas to solve it? – Crostul Jun 26 '17 at 11:19
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@Crostul, Yes I have the solution, because in our group every member should provide the solution of his designed problem. – Seyed Jun 26 '17 at 11:35
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@RobertZ, It involves small angles and therefore we need to use trigonometry. – Seyed Jun 26 '17 at 12:43
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@Seyed After some unpleasant calculation I found that the maximum volume is around $7.83415\pi$ (not a rational multiple of $\pi$.). – Robert Z Jun 26 '17 at 12:47
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@RobertZ, Excellent work, well done. Your answer is correct but I don't think the calculation is unpleasant. It is all depends on how you approach the problem. – Seyed Jun 26 '17 at 13:25
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@Seyed Is the answer supposed to be an approximation or the exact value? – Robert Z Jun 26 '17 at 13:28
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@RobertZ, It is not exact and your answer is correct. – Seyed Jun 26 '17 at 13:33
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@Seyed OK. Thanks for the problem. – Robert Z Jun 26 '17 at 13:35
2 Answers
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Here we need the the volume of a spherical cap: $$V=\frac{\pi h}{6}(3a^2+h^2)$$ where $a$ is the radius of the base of the cap and $h$ is the height of the cap. In order to find $h$ and $a$ we consider Cartesian coordinate system with the $x$-axis along the glass axis and with the origin at the center of the base of the glass. Take the circle of center $(5+4,0)$ and of radius $4$ and the tangent line through $(0,5/2)$. The water level is a line parallel to this tangent which passes through the point $(5+4+\sqrt{4^2-(6/2)^2},6/2)$.
After some unpleasant calculation I found that the maximum volume is around $7.83415\pi$ (not a rational multiple of $\pi$).
Robert Z
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Here is my solution to the maximum volume of water in a glass. The important step in this solution is to find the angle of rotation of the glass when it is resting on the table.
Seyed
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