Spheres in a Cylinder

Date: 05/20/2003 at 21:50:41
From: Juggy
Subject: Cylinders

Spheres of 6 cm radius are dropped into a cylinder of radius 8 cm and 
height 36 cm. How many spheres can fit into the cylinder? What is the 
distance from the top of a sphere to the top of the cylinder? How much 
free space is left in the cylinder?

Date: 05/21/2003 at 12:51:27
From: Doctor Peterson
Subject: Re: Cylinders

Hi, Juggy.

Let's look at how two spheres touch one another and the walls of the 
cylinder:

    |                                           |
    |                                           |
    |                    ooooooooooo            |
    |                oooo     |     oooo        |
    |             ooo         |         ooo     |
    |            o            |            o    |
    |          oo             |r            oo  |
    |         o               |               o |
    |        o                |                o|
    |        o                |                o|
    |       o                 |        r        o
    |       o                 +-----------------o
    |       o                 |                 o
    |        o               |:                o|
    |        o               |:                o|
    |         o            r| :               o |
    |          oo           | :             oo  |
    |            o         |  :            o    |
    |             ooo      |  :         ooo     |
    |                oooo |   :x    oooo        |
    |            ooooooooo|ooo:ooooo            |
    |        oooo         | oo:o                |
    |     ooo            |    : ooo             |
    |    o               |    :    o            |
    |  oo              r|     :     oo          |
    | o                 |     :       o         |
    |o                 |      :        o        |
    |o                 |      :        o        |
    o                 |       :         o       |
    o-----------------+.......:         o       |
    o        r        |  R-2r           o       |
    |o                |                o        |
    |o                |                o        |
    | o              r|               o         |
    |  oo             |             oo          |
    |    o            |            o            |
    |     ooo         |         ooo             |
    |        oooo     |     oooo                |
    +------------ooooo|ooooo--------------------+

Can you find x, the height of the right triangle I've drawn in? That 
will be useful for solving the problem.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/