Pyramid into fourths
(follow-up to "Cutting a triangle")
See also Cutting a triangle
Construct from poster board a pyramid with a square base having sides 10 cm long that has equilateral triangles for faces. On each face draw three horizontal lines marking three different levels, for 1/4 of the volume, 1/2 of the volume, and 3/4 of the volume of the pyramid.
Solution
Take as the linear measure of a pyramid the length x of an edge of its base (and its side). The volume V(x) is proportional to the cube of its linear measure, V(x) = c*x3, where c is a constant (which we could have computed if we wanted to).
To get the top 1/4 of the volume we need to find x such that,
V(x)/V(10) = 1/4
So c*x3/(c*103) = 1/4
therefore x = 10*3√(.25) = 6.3 cm. (3√ is the cube root)
To get the top 1/2 of the volume, we need to find x such that,
V(x)/V(10) = 1/2
So c*x3/(c*103) = 1/2
therefore x = 10*3√ (.5) = 7.9 cm.
To get the top 3/4 of the volume we need to find x such that,
V(x)/V(10) = 3/4
So c*x3/(c*103) = 3/4
therefore x = 10*3√ (.75) = 9.1 cm.
So we need to make three marks at each EDGE radiating from the apex of the pyramid at the distances 6.3 cm, 7.9 cm, and 9.1 cm from the apex, and to draw the lines between them before assembling the pyramid. It looks like this:
You can check your solution empirically using rice.