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# Hard Geometry Question

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Polyhedron "P" is inscribed in a sphere of radius 36 (meaning that all vertices of  lie on the sphere surface). What is the least upper bound on the ratio

$$\frac{\text{volume of }P}{\text{surface area of }P}~?$$

In other words, what is the smallest real number "t" such that:

$$\frac{\text{volume of }P}{\text{surface area of }P} \le t$$

must be true for all polyhedra that can be inscribed in a sphere of radius 36?

Jun 29, 2020

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Hi!

BUSTED!! this is an AoPS alcumus question

but here:

as volume increases, the ratio becomes greater. Think about what would make the volume very large. It could be large to the point where it could be barely contained in the sphere. You can try to find the approximate ratio now!

Nov 1, 2020