The Light At The End Of The Tunnel

Calculus Level 3

Suppose a light source is positioned in R2\textbf{R}^{2} at the origin (0,0).(0,0). Smooth mirrors are positioned along the lines y=0y = 0 and y=2y = 2 from x=0x = 0 to x=10.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 1515^{\circ} and 7575^{\circ} with the positive xx-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.x = 10.

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

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