A unit sphere is on a level surface in the rain. Find the largest of each of the following shapes that can be placed under the sphere and not get rained on:

a) Sphere
b) Cube
c) Regular tetrahedron

(a) I believe the radius of the sheltered sphere woud be:
6 - 4*sqrt(2).

(b) & (c) It seems to me that the two other sheltered polyhedra requires the use of calculus, to which I am too weak on, to find the answer. (b) Due to the curvature of the sheltering sphere and the straight edges of the cube, the cube must be located further underneath the circumference of the great circle that is parallel to the base plane. Approximating the side length of the sheltered cube to accomodate this need, I find the length is approximately 2/7 of the radius of the sheltering cube.

 

Edited on April 29, 2012, 3:12 am

Posted by Dej Mar on 2012-04-27 16:09:51