A mouse on a ring

Calculus Level 4

A mouse is running along a circle of radius 1, with center \((1,0)\) at a constant speed 1. A lazor is positioned at the origin, and pointed towards the mouse at all times. The mouse started running at the origin, at time \(t=0\).

Given that the angle the lazor makes with the horizontal (the blue line in the GIF) at time \(t\) is \(f(t)\), and that

\[\int _{ 0 }^{ \pi }{ f(t) } \ dt=\frac { A{ \pi }^{ B } }{ C } \]

Find the value of \(A+B+C\)

\(\bullet\) \(f(t)\) is in radians.

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