Three Spheres in a Dish

Date: 08/04/99 at 23:18:31
From: Kruger Guy
Subject: Radius and spheres

Hey you people are great! I always read through the archives, and I 
have sent in a few problems before. Now I'm a first-year university 
student and I still find Dr. Math really useful.

Anyway, I have this problem that's really bugging me. It was in the 
Australian Schools national maths competition, so it's supposed to be 
solved without a calculator.

Here's the problem:

3 spheres with radii of 10 cm are inside a hemispherical dish. It is 
noticed that the tops of all 3 spheres are exactly even with the top 
of the dish. What is the radius of the dish?

I tried to do a diagram but it didn't work. I hope you understand the 
problem. I'm absolutely stumped for a way to even start this problem, 
so any help would be greatly appreciated.

Thanks heaps and keep up the good work.

Kruger

Date: 08/05/99 at 12:15:22
From: Doctor Peterson
Subject: Re: Radius and spheres

Hi, Kruger.

A top view will look like this:

                    **********
              ******          ******
           ***                      ***
        ***ooooooooo           ooooooooo**
      **ooo         ooo     ooo         ooo*
     * o               o   o               o*
    * o                 o o                 o*
  ** o                   o                   o**
 *   o         +-------------------+         o  *
 *   o          \        o        /          o  *
*     o          \      o o      /          o    *
*      o          \    o   o    /          o     *
*       ooo        \ oooo*oooo /        ooo      *
*          oooooooooo\   |   /oooooooooo         *
*                o    \  |  /    o               *
 *              o      \ | /      o             *
 *             o        \|/        o            *
  **           o         +         o          **
    *          o                   o         *
     *          o                 o         *
      **         o               o        **
        ***       ooo         ooo      ***
           ***       ooooooooo       **
              ******            ****
                    **********

The equilateral triangle has sides 2r, where r = 10 cm is the radius 
of the small spheres. The distance from the center of any of these 
spheres to the center of the larger sphere will be 2/3 of the altitude 
of the triangle.

A side view cutting through one of the small spheres will look like 
this:

             R
*-------ooooooooo--------+----------------------*
*    ooo         ooo  /  |                      *
*   o               +    |r                     *
*  o           r /   o   |                      *
 *o           /       o  |                     *
 *o         +------------+                     *
  *      /            o                       *
   *   / r           o                       *
    +               o                       *
     *oo         ooo                       *
      * ooooooooo                         *
       *                                 *
        **                             **
          **                         **
            ***                   ***
               ******       ******
                     *******

The horizontal segment D is the same as D just calculated. See 
where you can go from here.

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