Optimisation...Sphere inside a Cone Problem

[dr|ft]

A sphere of radius 8 cm is circumscribed by a right circular cone. If the cone is to have a max volume, find the height of the cone and the radius of the base of teh cone...

 

I need to relate the radius of the sphere to either the radius of the base or height of the cone, i can't find anything, any ideas people?

 

Vol of Sphere = 4/3 TTr^3

Vol of Cone = 1/3 .TT.r^2.h

[mezarashi]

Alright, I'm a bit confused, but by your use of "circumscribed by", you mean that the cone is inside the sphere? Because if the sphere is inside the cone, and no height dimensions are given then the maximum volume is infinity. And since circumscribed usually refers to the smallest possible fit, trying to find the max of a min is a bit weird.

[dr|ft]

ahh it must be cone inside a sphere then, thanks for that

[mezarashi]

Okay, in that case, this is going to be a calculus exercise. The easiest way to find the relationship is by taking the cross-section of the sphere and cone. Then you are back to two-dimensions on paper

 

If it's any clearer, you can draw the cone upside down so that the base will be on the upper hemisphere of the sphere. Have the x-y axis running through the center of the sphere. Then increasing the angle from the x-axis you can see the base of the cone getting smaller while the height increases.

 

The radius of the cone at any point is simply x = Rcos a, where R is the radius of the sphere and a is the angle. The height at any point is y = Rsin a (+ R, the other side of the sphere).

 

Using the volume equation: 1/3 *pi * r^2 *h, plug everything and differentiate for the first maxima.

[dr|ft]

thanks a lot i got it all figured