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 Question from Neven: The cone of smallest possible volume is circumscribed about a given hemisphere. What is the ratio of its height to the diameter of its base? (G.F.Simmons, Calculus with Analytic Geometry, CH4 Applications of Derivatives)

Neven,

I assume that your cone is a right circular cone. In other words, a cross section containing the axis is an isosceles triangle with its incircle. Assume that the radius of the sphere (and therefore of the incircle) is 1. Let x be the distance from the vertex of the cone to the centre of the sphere, t be the length of a tangent line from the vertex, and r be the radius of the base of the cylinder.

Using similar triangles and Pythagoras theorem write the volume of the cone as a function of x. Use calculus to find the value of x that minimizes the volume of the cone and the ratio you want is (x + 1)/(2r).

Chris

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