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MATH / 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 …

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problem

Polyhedron PP is inscribed in a sphere of radius 3636 (meaning that all vertices of PP 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 tt such that volume of Psurface area of P≤t\frac{\text{volume of }P}{\text{surface area of }P} \le tmust be true for all polyhedra PP that can be inscribed in a sphere of radius 3636?
Plain-text mathematical notation (without MathML)
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 (volume of P)/(surface area of P)≤tmust be true for all polyhedra P that can be inscribed in a sphere of radius 36?
Original LaTeX notation
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 $36$?

level

Level 5

type

Geometry

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Source and history

task_id
51672c08-28fb-5ca5-a053-e4350c63a6fa
task_key
geometry--test--51672c08-28fb-5ca5-a053-e4350c63a6fa
task_revision_id
1
config / split
geometry / test

Official source

initial import