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

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The discussion revolves around determining the height of a cone confined within a hemisphere of radius r, with the cone's base aligned with the hemisphere's base. The proposed solution suggests that the height of the cone should equal the radius of the hemisphere, leading to confusion about the correct relationship between the two. Participants emphasize the need to apply calculus, specifically derivatives, to find the minimum volume of the cone, indicating that the problem involves complex mathematical functions. There is also a warning against sharing personal information like email addresses in public forums. Ultimately, the conversation highlights the importance of correctly understanding geometric relationships and the application of mathematical principles to solve the problem.
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
 
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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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