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 re: some thoughts (Clarification.) by Jer)

I will then assume you meant the width of the rectangle to be wide enough to shelter the polyhdedra/sphere given that the angle it is proped and its length are sufficient otherwise.

The terms "long rectangle", "long board" and "long prism" had not really been defined. To me, "long" in the context it is used denotes that the length or width is at least twice that of the width or length, respectively. Yet that is only an assumption.  
The adjective did not define the direction of the length or width of the board - though your clarification has. 

Posted by Dej Mar on 2012-05-03 15:29:33