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 3-D Pythagorean theorem for the maximal spheres is
(8.5-2r)²+(11-2r)²+(6.25-r)²=(2r)²
Which simplifies to the quadratic
5r²-91.5r+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 2011-07-11 02:02:46