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Sphere-Cone (Posted on 2016-12-25) Difficulty: 3 of 5

Given a sphere of fixed radius a. A right circular cone is is to be
found which will enclose the sphere such that the sphere is tangent
to the cone's lateral surface, the sphere is tangent to the cone's
base at its center, and the ratio Ac/As is minimised where Ac is the
cone's surface area (both lateral and base) and As the sphere's
surface area.

To keep solutions uniform let's denote the cone's altitude by h, base
radius by r and slant height by L.

Also, give the minimised ratio.

See The Solution Submitted by Bractals    
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Some Thoughts Possible solution | Comment 1 of 2

Ac/As = 1/4(1+2^(1/2))^2, around 1.4571.

The above is not correct, because it does not add in the term for the base correctly. When the base is included, the ratio of the two areas Ac/As is 2:1 - see Brian's answer.

Edited on December 26, 2016, 1:26 am
  Posted by broll on 2016-12-25 08:56:05

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