It is the classical will-my-cables-fit-within-the-tube-problem which lead me to the interest of circle packing. So basically, I have 3 circles where r = 3 and 1 circle where r = 7 and I am trying to find the least r for an outer circles of these 4 smaller circles.
After a couple of hours of thinking and some sketches with a compass I am getting close to the actual result.
But how can I calculate this?
With what formula?
EDIT:
Thanks for the great answer.
And then I come to wonder.
What happens if you add another of the small circles, so you have four circles with r = 3? It is very close to 11.7
Best Answer
Clearly radius 10 is not quite enough; once you have an arrangement like this, it may be possible to find the eaxct outer radius, but in any case it can be estimated fairly well. Reminds me of these, http://en.wikipedia.org/wiki/Sangaku
I drew in some trial outer circles, $r = 11$ worked with room to spare, so I split the difference, $r=10.5 = 21/2$ also worked with just a little extra room.
EDIT: did it in coordinates, I thought it was going to be a degree four polynomial but there was cancellation and it became linear, the best outer radius is $$\frac{637}{61} \approx 10.4426 $$
EEDDIITT: did it over with symbols. If the larger given radius, now 7, is called $A,$ and the smaller given radius, now 3, is called $B,$ then the radius of the circumscribed circle is $$ R = \frac{A^2 (A+2B)}{A^2 + AB - B^2}. $$