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
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