+85 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 volume of Psurface area of P ? In other words, what is the smallest real number t such that volume of Psurface area of P≤t must be true for all polyhedra P that can be inscribed in a sphere of radius 36?
+179 So for this question, formulas for volume and surface areas of spheres are essential. The volume of P is defined as 43πr3=43π363, and Surface area of P is 4πr2, which in this case is equivalent to 4π362. Once we put this into the inequality (volume of P)/(surface area of P) ≤ t, we have, after simplification 12≤t. This, the smallest real number t should be 12.
