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 the width of the rectangle is given as 1, the largest of each shape that can be sheltered is:
Sphere -> diameter = 1
Cube -> edge length = 1
Regular tetrahedron -> edge length = 8 - 4*sqrt(3)

The minimal length of the rectangle to shelter the sphere will depend on the angle formed, and vice versa. In order to shelter the polyhedra the angle formed must be between 0 and pi/4
radians, exclusive. The angle that permits the minimal length of the rectangle to shelter the polyhedra will be pi/8 radians.
As the angle approaches 0 or pi/4 radians the length of the rectangle will approach infinity.

 

 

Posted by Dej Mar on 2012-05-03 13:21:49