In this diagram semi-circles are constructed on diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$, so that they are mutually tangent.  If $\overline{CD} \bot \overline{AB}$, then the ratio of the shaded area to the area of a circle with $\overline{CD}$ as radius is:
$\textbf{(A)}\ 1:2\qquad \textbf{(B)}\ 1:3\qquad \textbf{(C)}\ \sqrt{3}:7\qquad \textbf{(D)}\ 1:4\qquad \textbf{(E)}\ \sqrt{2}:6$
To make the problem much simpler while staying in the constraints of the problem, position point $C$ halfway between $A$ and $B$. Then, call $\overline{AC} = \overline{BC}=r$ . The area of the shaded region is then \[\frac{ \pi r^2 - \pi (r/2)^2 - \pi (r/2)^2}{2}=\frac{\pi r^2}{4}\]
Because $\overline{CD}=r$ the area of the circle with $\overline{CD}$ as radius is $\pi r^2$.
Our ratio is then \[\frac{\pi r^2}{4} : \pi r^2 = 1:4\]
Which corresponds with answer $\fbox{D}$