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 20120503 13:56:42 