Interesting challenging Problem

[maxhk]

Hello Everyone,
I am trying to solve this difficult problem :

"A heavy spherical ball is lowered carefully into a full conical wine
glass whose depth is \(\displaystyle h\) and whose generating angle (between the axis
and a generator) is \(\displaystyle \alpha\). Show that the greatest overflow occurs when
the radius of the ball is \(\displaystyle \dfrac{h\ \sin(\alpha)}{\sin(\alpha)+\cos(2\alpha)}\)".

Work:

1- If the ball is fully immersed, I got
Radius of the ball \(\displaystyle = \dfrac{h\ \sin(\alpha)}{1+\sin(\alpha)}\)

2- The ball may be too large to be fully immersed and
yet displace more liquid than a smaller ball that is fully immersed.

How to find it in this case ?
Any ideas ? Thanks

 

[tkhunny]

You will need the volume of the original cone and the volume of your sphere. You may need a third case.

1) Fully submerged.
2) More than 1/2 submerged.
3) Less than 1/2 submerged.

 

[Deleted member 4993]

Hello Everyone,
I am trying to solve this difficult problem :

"A heavy spherical ball is lowered carefully into a full conical wine
glass whose depth is \(\displaystyle h\) and whose generating angle (between the axis
and a generator) is \(\displaystyle \alpha\). Show that the greatest overflow occurs when
the radius of the ball is \(\displaystyle \dfrac{h\ \sin(\alpha)}{\sin(\alpha)+\cos(2\alpha)}\)".

Work:

1- If the ball is fully immersed, I got
Radius of the ball \(\displaystyle = \dfrac{h\ \sin(\alpha)}{1+\sin(\alpha)}\)

2- The ball may be too large to be fully immersed and
yet displace more liquid than a smaller ball that is fully immersed.

How to find it in this case ?
Any ideas ? Thanks

This is problem of olive sinking into martini.

Anyway - to start-off you need to consider two cases:

1) The diameter of the sphere submerged inside the cone.

2) The diameter of the sphere is outside the cone.

 

[maxhk]

Thanks for you both!
I will try your suggestions ...