Date: Dec 16, 2012 7:15 AM
Author: Brian Chandler
Subject: Re: Missouri State University Problem Corner
Michael Press wrote:

> In article <Pine.NEB.4.64.1212090027020.10188@panix1.panix.com>,

> William Elliot <marsh@panix.com> wrote:

>

> > > Three unit spheres are mutually tangent to one another and to a

> > > hemisphere, both along the spherical part of the hemisphere and

> > > along its equatorial plane. Find the radius of the hemisphere.

All those equations and things made my head spin. Here's a simpler

calculation.

Three unit spheres touch; therefore their centres (S1, S2, S3) are at

the vertices of an equilateral triangle of side 2. They also sit on

the flat face of the hemisphere; so the height of each of the centres

over the flat face is 1, and symmetry implies that the centre (C) of

the hemisphere is under the centre (T) of the equilateral triangle,

and the centres S1, S2, S3, and C form a pyramid of height 1 on a

base of side 2.

Since each sphere touches the curved surface of the hemisphere, the

normal to the point of contact goes through the centre of the sphere

and the centre of the flat face. Therefore the radius of the

hemisphere is the length of a sloping edge of the pyramid (e) plus the

radius of a sphere (1).

(Using r() for square root...)

Two applications of Pythagoras' theorem give us the distance from the

centre of the triangle to a vertex:

d = 2 / r(3) (30-60-90 triangle; longer right side = 1)

And sloping edge

e = r( d^2 + 1^2) = r( 4/3 + 1) = r(7/3)

So radius of hemisphere is 1 + r(7/3)

> That looks large. I get a different answer.

Hmm.

> Let r denote the radius of the hemisphere.

> Label the center of the hemisphere O.

There's a real question: what position _is_ the centre of a

hemisphere...

Brian Chandler