This is in continuation of
Two circles.
In a 8.5x11x13.5 hollow
rectangular cuboid, I place
three identical solid spheres of equal volume  all completely inside the cuboid, of course.
What's the largest portion of the cuboid (in terms of volume) that these spheres can contain?
What would be the answer if I placed FOUR equal spheres?
First to describe the configuration: Call one of the 11x13.5 rectangles ABCD, then EFGH is the corresponding opposite rectangle.
Tuck one sphere into the corners A and B and the third along the midpoint of GH.
The 3D Pythagorean theorem for the maximal spheres is
(8.52r)²+(112r)²+(6.25r)²=(2r)²
Which simplifies to the quadratic
5r²91.5r+238.8125 = 0
Whose discriminant is 3596 and approximate solution is r=3.1533426
I won't bother with the portion of the cuboid filled. If this is the largest r it will yield the largest proportion.

Posted by Jer
on 20110711 02:02:46 