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]

Suppose the martini glass has a sidelength $L$ and top radius $r$ (see figure below). We’ll compare the volume of liquid in both configurations, which will give us the desired relationship between $p$ and $q$. It turns out we can use a simple proportionality argument without needing to actually compute either volume.

In the configuration on the left, the liquid cone is a shape similar to that of the entire glass, but with each dimension scaled by $p/L$. Therefore the volume of the liquid is proportional to $p^3$.

Moving forward, we’ll need to know a bit about conic sections. Namely, the intersection of a plane with a cone is always either an ellipse, a hyperbola, or a parabola. You can learn more about conic sections here.

Now consider the configuration on the right. The volume of the liquid cone is $\frac13 \times (\text{area of liquid surface})\times (\text{height})$. The liquid surface is a conic section (an ellipse), which has area $\pi s w$, where $s$ and $w$ are the major and minor radii respectively. Rewrite the volume as: $\frac{\pi}{3} \times w \times (s \times \text{height} )$.

The product $(s \times \text{height})$ is simply the lateral area of the liquid when viewed from the side. This triangle has one side $L$, one side $q$, and the angle between them (the tip of the glass) is fixed. Therefore this lateral liquid area is proportional to $ q$.

The minor radius $w$ belongs to a plane that passes through the midpoint of the liquid surface as well as the midpoint of the top opening of the glass (with radius $r$). By similar triangles, this slice is parallel to the opposite edge of the glass so the conic section is a parabola this time. It follows that $w$ is proportional to $\sqrt{q}$.

Putting everything together, the volume of the liquid on the right is proportional to $q \times \sqrt{q} = q^{3/2}$. The volumes of liquid in both configurations are equal so we conclude that $p^3 \propto q^{3/2}$. Or, in other words, $q \propto p^2$. To find the constant of proportionality, observe that when $p=L$, we also have $q=L$. Therefore:

\[

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

\]

So when you tip the glass to the point of overflowing, the drink reaches up the opposite side of the glass a fraction equal to the square of the fraction reached when the glass is level.

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!

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?

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!

Great, thank you.