What is the Height of a Cone Confined to a Hemisphere with Given Radius?

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Discussion Overview

The discussion revolves around determining the height of a cone that is confined within a hemisphere of a given radius. The focus is on mathematical reasoning, particularly involving optimization and geometric relationships between the cone and the hemisphere.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant claims that the height of the cone is H = (sqrt of 3) R, where R is the radius of the hemisphere.
  • Another participant suggests that if the bases of the cone and hemisphere are in the same plane, the height of the cone should equal the radius of the hemisphere, proposing that the vertex of the cone is at the base of the hemisphere.
  • A participant mentions the need for derivative applications to find maximum and minimum values, indicating uncertainty about the specific functions to use, possibly involving the volume of the cone.
  • There is a reiteration of the derivative approach, with a participant expressing difficulty in identifying the necessary functions, while also noting the complexity of the problem.
  • A later reply questions the validity of the initial claim regarding the height of the cone being greater than the radius of the hemisphere, suggesting that such a result is impossible.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between the height of the cone and the radius of the hemisphere, with no consensus reached on the correct height or the methods to derive it.

Contextual Notes

Participants have not clearly defined the objective function or the specific geometric relationships, leading to potential ambiguities in the problem setup. The discussion reflects a lack of resolution regarding the mathematical steps involved in deriving the height of the cone.

leprofece
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(1) Confine to a hemisphere of RADIUS r a volume minimum Cone; the plane of the base of the cone matches with the basis of the hemisphere. Find the height of the cone.

Answer is H = (sqrt of 3) R
 
Last edited:
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I think if the two objects have their bases in the same plane, the the height of the cone must be the same as the radius of the hemisphere, so I think more likely the case is that the vertex of the cone on on the base of the hemisphere. Even so I get a result that is very similar to, but critically different than what you have given.

1.) What is your objective function?

2.) Can you express the radius of the cone in terms of the radius of the hemisphere and the height of the cone? Look at a cross-section through the center of both objects and Pythagoras will be your friend.

What do you find?
 
Ok it is a problem of derivative applications Maximum and minimum we must get two functions and derive one to get the minimum.
In this problem I don't have idea of the functions maybe the volume of the cone and thales
as i said before they are very difficult problems.
 
leprofece said:
Ok it is a problem of derivative applications Maximum and minimum we must get two functions and derive one to get the minimum.
In this problem I don't have idea of the functions maybe the volume of the cone and thales
as i said before they are very difficult problems.

I edited your post to remove your email address. It is not a good idea to publicly post your email addy and it is not our policy to email solutions anyway. It is best to respond in the threads.

Did you try what I suggested? It really makes the problem fall into place. :D

Do you see how the result you posted is impossible? How can the height of the cone be greater than the radius of the hemisphere?
 

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