This is a question from the 2015 AMC12 math competition. I haven't really made much progress at all on it, and I just want to know the right way to solve this equation.
A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k \ge 1$, the circles in $\bigcup_{j=0}^{k-1}$ are ordered according to their points of tangency with the $x$-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer $L_k$ consists of the $2^{k-1}$ circles constructed in this way. Let $S = \bigcup^6_{j=0} L_j$, and for every circle $C$ denote by $r(C)$ its radius. What is $$\sum_{C\in S}\frac1{\sqrt{r(C)}}?$$
$\mathbf{(\ A\ )} \frac{286}{35}$
$\mathbf{(\ B\ )} \frac{583}{70}$
$\mathbf{(\ C\ )} \frac{715}{73}$
$\mathbf{(\ D\ )} \frac{143}{14}$
$\mathbf{(\ E\ )} \frac{1573}{146}$
There is also an illustration of the problem - it provides no new information, just a way to visualize it.