2017 NS 30

\(\text{Let }p \text{ be the point of intersection}.\\ \text{The angle a line from }(0,1) \to p \text{ makes with the y axis is }\\ \theta = \dfrac{\pi}{6}, \text{ so }p =\left (\sin\left(\dfrac \pi 6\right),~1-\cos\left(\dfrac \pi 6\right)\right)\\ p = \left(\dfrac 1 2, \dfrac{2-\sqrt{3}}{2}\right)\\ \text{Let }r \text{ be the radius of the smaller semicircle}\\ r^2 = \|p-(r,0)\|^2\\ r^2 =\left(\dfrac 1 2 - r\right)^2+\left(\dfrac{2-\sqrt{3}}{2}\right)^2\\ \text{Solve that for }r\)