Solving Sphere in Cone Problem w/Semivertical Angle

[PhysicsInept]

Help please - okay so I have a question and struggling here.

I need to know the radius of the sphere and how much water it displaces.
One sphere inside an inverted cone
One sphere for which the maximum possible amount of water is displaced.

The problem is I’m only given the height of the cone and the semivertical angle Any idea what formula to use? I found some but none account for semivertical angle.

Thanks

 

[anuttarasammyak]

Cut the cone half and get the triangle cross section. You observe in the triangle the relation
of height h, semivertical angle $\phi$ and radius a of the sphere.

 

[PhysicsInept]

Help please - okay so I have a question and struggling here.

I need to know the radius of the sphere and how much water it displaces.
One sphere inside an inverted cone
One sphere for which the maximum possible amount of water is displaced.

The problem is I’m only given the height of the cone and the semivertical angleAny idea what formula to use? I found some but none account for semivertical angle.

Thanks

Cut the cone half and get the triangle cross section. You observe in the triangle the relation
of height h, semivertical angle $\phi$ and radius a of the sphere.

 

[PhysicsInept]

Thanks. So what do I do if I don’t know the radius?

 

[PhysicsInept]

For example, let’s say the height of the cone is 1.5m and the semi vertical angle is 55 degrees?

 

[anuttarasammyak]

Why do not you draw the triangle rough sketch and put a, h and $\phi$ into it to find a general relation.

 

[Lnewqban]

Welcome to PF!

See how to do it graphically, if interested:
https://www.mathsisfun.com/geometry/construct-triangleinscribe.html

See how to calculate it using trigonometry:
https://en.wikipedia.org/wiki/Law_of_cotangents

https://math.libretexts.org/Bookshe...les/2.05:_Circumscribed_and_Inscribed_Circles

 

[PhysicsInept]

Why do not you draw the triangle rough sketch and put a, h and $\phi$ into it to find a general relation.

But we don’t know if the sphere is fully submerged in the water (in the cone) or part of it rests above water. Therefore the radius of the sphere wouldn’t be determined based on the cone triangle calculation?

 

[anuttarasammyak]

In the condition that the sphere submerges completely you may agree that the maximum size of a exists. You should get it first.

 

[mfb]

I moved the thread to our homework section.

Find the position and then the displacement as function of the radius, then find the radius where this displacement becomes maximal.

 

[Orodruin]

In the condition that the sphere submerges completely you may agree that the maximum size of a exists. You should get it first.

Regardless of this condition a maximum will exist. For the general case, both possibilities must be considered. However, the fully submerged case is relatively simple.

 

[Lnewqban]

But we don’t know if the sphere is fully submerged in the water (in the cone) or part of it rests above water. Therefore the radius of the sphere wouldn’t be determined based on the cone triangle calculation?

Can you post the entire text of the problem?

 

[PhysicsInept]

​

A infinite herd of perfectly spherical cows of every possible real-valued radius is passing by a perfectly conical lake in single file line. Each cow swims in the lake and displaces a certain amount of water once it comes to rest flush against the sides of the lake. Depending on the radius, the cow could be entirely submerged, or partially submerged.​

There is a single cow for which the maximum possible amount of water is displaced. The question is, what radius does this cow have and how much water does this cow displace?
This conical lake is filled to the top with a depth of exactly 1.4914 meters from the vertex to the surface. The lake has a semivertical angle of exactly 61.692 degrees.NOTE: This is not homework for school. It is part of a puzzle for a hobby I’m in to.
Here is what I’ve done so far to at least hopefully determine the radius of the sphere (cow) but that would only be true if exactly half submerged?
I am NOT a physics person lol. It’s been 20 years since high school!

 

[anuttarasammyak]

I made sketches for your help. By increasing radius of sphere
Fig1 From radius zero to maximum radius of fully submerge condition
Fig2 Between Fig 1 and Fig 3
Fig3 From the sphere begins to touch at the base circle of the cone to radius infinity

Getting radius of two solid line spheres would help you.