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### Three Spheres in a Dish

```
Date: 08/04/99 at 23:18:31
From: Kruger Guy

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

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/
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Higher-Dimensional Geometry

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