# The Light At The End Of The Tunnel

Calculus Level 3

Suppose a light source is positioned in $$\textbf{R}^{2}$$ at the origin $$(0,0).$$ Smooth mirrors are positioned along the lines $$y = 0$$ and $$y = 2$$ from $$x = 0$$ to $$x = 10.$$ The light source is then oriented so that the beam of light emanating from it makes an angle, chosen uniformly and at random, of between $$15^{\circ}$$ and $$75^{\circ}$$ with the positive $$x$$-axis. The beam is then allowed to reflect back and forth between the two mirrors until it "exits the tunnel", that is, it crosses the line $$x = 10.$$

If the expected distance that the beam of light travels from its source until it exits the tunnel can be expressed as $$\dfrac{10}{\pi}\ln(a + b\sqrt{c}),$$ where $$a,b,c$$ are positive integers with $$c$$ square-free, then find $$a + b + c.$$

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