@Tom fun stuff. But seriously, let’s unpack the ellipsis. How do we put that question to a class, aiming at clarity and concision? I’m thinking, “if you fill the club soda up to the 6 cm marker, how high will the drink go after you add the cranberry juice?” Which I dig only because it’d be fun to talk about the obvious wrong answer.

@Sue, I think that’s what Jason means when he suggests a geometric solution, though California certainly doesn’t include analytic geometry of that rigor in its standards.

PS. 16.5 cm is extremely accurate, keeping in mind that we start measuring the soda 3 cm up the scale. Here is the answer photo. (Does it go without saying that WCYDWT media-based questioning is better when you have “answer media” that students can contrast with their own work?)

eh?

When I read “do you see where this is going? How we’re (quickly) going to rule out Geometry?” I first thought you meant this one couldn’t be solved without calculus. But considering the glasses you bought the other day, I now see that you mean there will be other how-high problems where we no longer have a cone.

Anyway, here’s me no-calc solution:
In your equation, y = 1/12 x +1.75, the x is the height and the y is the radius. It seemed easier to me to imagine the whole cone involved (go down or back to the point), so I found the x-intercept of this line, which is x= -21, and then said from there the radius is always 1/12 the height.

V=pi/3 * r^2*h. But r=1/12 h, so V = pi/432 * h^3.

But the bottom part can’t have liquid, so
V =354.88 =pi/432*h^3-pi/432*21^3.
Soling for h gives 37.5, and subtracting 21 to get back to the original problem gives 16.5cm, which is not to the top.

Does this count as geometry?

NB. My calculus, apparently, sucks. I’d expect some error due to estimation, the thickness of the glass, etc., but my margin of error is on the order of 600%. Can someone check my work?