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Polyhedron P is inscribed in a sphere of radius \(36\) (meaning that all vertices of P 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 \(P\) that can be inscribed in a sphere of radius \(36\)?

 Dec 30, 2020
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Using variational calculus, the answer works out to 18.

 Dec 31, 2020

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