Biggest splash 2

[bonanova]

If a sphere is dropped into a conical goblet filled with water, there is a sphere radius for which a maximal fraction of water is displaced (spilled) from the goblet. The maximum fraction varies with the base angle of the goblet. What is the greatest value of the maximum fraction? For what base angle does it occur?

[Logophobic]

Are you sure about that?

Spoiler

We have already the maximum displaced volume (V_d)for any given cone, and we know the volume of the cone (V_c), so you are for the maximum V_d/V_c as a function of θ?

Well, without reproducing the formulas for V_d and V_c here, and without working out the proof, I am going to suggest that, for θ = BaseAngle/2, we have f(θ) = V_d/V_c with the domain 0 < θ < 90, and that this function is strictly increasing with a limit of 8/9 as θ approaches 90.

 

Edited December 18, 2015 by Logophobic

[bonanova]

You answered the question correctly; I'm not sure which part of it you're questioning. It's probably the fact that

Spoiler

There is no angle after which f(θ) decreases.

There's a bit of a paradox associated with the fact that. when θ = 90, V_c = 0, and so therefore is V_d.
Did you need l'Hôpital's rule?.