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.

Part 1

Noting that the vertical cross section of the shaded space is a right angled triangle with base cos(theta) and height sin(theta) and, in each case, using a right angled triangle in this vertical section....

(a) Sphere, radius r       tan(theta/2) = r/(cos(theta) – r)

            which gives:      r = cos(theta)/(1 + cot(theta/2))

(b) Cube, side a             tan(theta) = a/(cos(theta) – a)

            which gives:      a = sin(theta)/(1 + tan(theta))

(c) Regular tetrahedron, edge b

Since its height is b*sqrt(2/3),     sin(theta) = a*sqrt(2/3)/cos(theta)

            which  gives:     b = (1/4)*sqrt(6)*sin(2*theta)

Bonus

I agree with Dej Mar’s answers and his statement that all three shapes are maximised when theta = Pi/4

Posted by Harry on 2012-05-06 11:20:12