Overflowing martini glass

This Riddler puzzle is all about conic sections.

You’ve kicked your feet up and have drunk enough of your martini that, when the conical glass (🍸) is upright, the drink reaches some fraction p of the way up its side. When tipped down on one side, just to the point of overflowing, how far does the drink reach up the opposite side?

Here is my solution:
[Show Solution]

7 thoughts on “Overflowing martini glass”

  1. Really beautifully executed. I really like the fact that you post cute problems as well on this blog I liked to bijection in the earlier post. Thank you for doing this!

  2. Very nice. But shouldn’t more should be said to justify the assumption that the constant of proportionality in the case of volume is identical to, instead of related to in some other way, the one in the case of the parabola?

    1. Ultimately, we end up proving that $q$ is proportional to $p^2$. This means there exists some constant of proportionality $C$ such that $q = C p^2$.

      The key part here is that $ C$ is a constant; it doesn’t depend on $p$ or $q$. So we can solve for $C$ by just picking particular values of $p$ and $q$. We know that when $p=L$, we must have $q=L$, so the only possible choice for $C$ is $1/L$. This means the equation above becomes:

      $$\left( \frac{q}{L} \right) = \left( \frac{p}{L} \right)^2$$

      It’s also possible to calculate the exact constants of proportionality at every step in the proof and then multiply them out and obtain the same result, but that requires a lot more algebra!

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