A shelter is in a the shape of a long rectangle of width 1. The long side forms an angle of θ with the level solid ground. What is the largest of each of the following shapes that can be sheltered from the rain (as a function of θ)?

a) Sphere
b) Cube
c) Regular tetrahedron

As a bonus, for each shape, find the angle that maximizes it.

As to how to formulate a function of theta when there is only one value for theta to provide the largest shape to be sheltered from the rain, I am at a loss.  In all cases the angle that maximizes the shelter is 45-degrees (pi/4 radians) bevel to the solid level ground.  
(a) The largest sphere to be sheltered can have a radius of:
[sqrt(2) - 1]/2 units in length.
(b) The largest cube to be sheltered can have an edge length of: sqrt(2)/4 units.
(c) The largest regular tetrahedron to be sheltered, defying any gravitational force, will have a face of the tetrahedron tangent to the plane of the rectangle with its opposite vertex touching the ground (the vertex being directly beneath the top edge of the rectangle.) The tetrahedron's edge length will be:
sqrt(6)/4 units.

Edited on May 5, 2012, 9:08 am

Posted by Dej Mar on 2012-05-04 03:26:06