A right circular cone of radius R and altitude H is circumscribed about a sphere of radius r. Find the relation between H and rif the volume

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A right circular cone of radius R and altitude H is circumscribed about a sphere of radius r. Find the relation between H and rif the volume of the cone is to be minimum?

Answer:

The cone is circumscribed about a sphere meaning that the sphere is inside the cone and it touches the cone in one point and in one circle.

Volume of the cone:
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What we want to do is to express R as a function of r and H. Then, we can substitute that into the volume formula and minimize it with respect to H.

If we take a close look at the cross section of this geometric model, we obtain an isosceles triangle with inscribed circle. Circle has the radius r and triangle has base 2R and height H

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