Chapter 5.5, Problem 42E

Soap Bubbles When two bubbles cling together in midair, their common surface is part of a sphere whose center $D$ lies on the line passing through the centers of the bubbles (see the figure). Also $\angle ACB$ and $\angle ACD$ each have measure $60°$.

(a) Show that the radius $r$ of the common face is given by $r=\frac{ab}{a-b}$

[Hint: Use the Law of Sines together with the fact that an angle $\theta$ and its supplement $180°-\theta$ have the same sign.]

(b) Find the radius of the common face if the radii of the bubbles are 4 cm and 3 cm.

(c) What shape does the common face take if the two bubbles have equal radii?