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.

(In reply to some thoughts by Dej Mar)

Take a long board.  Call its length 1 unit.  Put it on the ground and prop up one edge.  One long edge of the board lies along the ground.   The angle can actually vary from 0 to 90 degrees (0 to pi/2 radians) but at the extremes there will be no shelter.

Basically you are fitting a shape into a long prism whose cross section is a right triangle with hypotenuse 1.

Posted by Jer on 2012-05-03 13:56:42