Okay, solution time:
Last week's puzzle was:
Suppose you have an equilateral triangle. The area of the triangle is exactly 1200 square centimetres. Now suppose you have twenty of those triangles. It's possible to assemble those twenty triangles into a closed three-dimensional shape, a regular polyhedron.
What would be the volume, in cubic centimetres, of the largest sphere that could fit inside the shape?
Web searches made this pretty easy:
20-sided polyhedron = icosahedron
Wikipedia has the icosahedron formulas we need:
Let a be the length of one edge of the icosahedron (one side of the initial equilateral triangle), then the radius r of the inscribed sphere is:
r = sqrt(3) * (3 + sqrt(5)) * a /12.
The volume V of a sphere (Wikipedia is a good source again) is:
V = (4/3) * PI * r ^ 3.
The triangle is equilateral, so from
http://www.mathwords.com/a/area_equilateral_triangle.htm, we get the area A of the triangle:
A = sqrt(3) * a^2 / 4.
now solving and substituting, we get:
a = sqrt(4 * A / sqrt(3))
r = sqrt(3) * (3 + sqrt(5)) * sqrt(4 * A / sqrt(3)) / 12
V = (4/3) * PI * [sqrt(3) * (3 + sqrt(5)) * sqrt(4 * A / sqrt(3)) / 12] ^ 3
Plugging in the orginal area of A = 1200 cm^2, we get V = 263793 cm^3.
But, no one put that answer, did they? lol
