Volume of a part of the cone

[ValeryOchkov]

Sorry,

I cannot find the formula for a volume of a part of the right circle cone (a small part or a large one). The plane is vertical (one branch of hyperbola in the section). V(r, h, H) := ???

 

[Fred_Kohlhepp]

Maybe:

[ValeryOchkov]

Thanks, Fred.

Today I have saw in my dream that a volume (of the part) of the cone is 1/3 of a volume (of the part) of the cylinder:

V(r, H, h) := ???

[ValeryOchkov]

Or it was not good dream - кошмар - le cauchemar?

[ValeryOchkov · Accepted solution]

Утро вечера мудренее! it is a Cone hoof!

See cyclowiki.org

[ValeryOchkov]

One problem for my students

[Werner_E]

Better use integrals instead of ready made forumulas from the net

Either that way

or better, because more generic and speedier, that way

BTW, the far too large picture you use to insert in your textx spoil the display of the thread on some displays.

It would be far more clear to insert smaller pics!

[ValeryOchkov]

Thanks, Werner! It is a fine solution!

This task can have an interesting continuation. The figure below shows three projections of a reservoir with a liquid at the bottom: a side view, a top view and an end view. First, it is required to calculate the area of the liquid mirror depending on the value of h, and secondly, to create an animation of the shape change of this mirror when the tank is turned from horizontal to vertical position. Hyperbola is first transformed into two intersecting straight lines, then again returns to the hyperbole, then turns into a parabola, an ellipse, and finally a circle...

[Werner_E]

Good luck for creating that animation!

[ValeryOchkov]

Some history - from Austria

Kepler Fass