If O is the nearest point on the shore to the lighthouse (L) then OL is perpendicular to the shore. If X is 2 miles from O along the shoreline, and angle θ=angle XLO, tanθ=OX/OL=2.
Let x=OX, then x=tanθ for X anywhere along the shoreline, and dx/dθ=sec²θ, which can be written dx=sec²θdθ. The rate of change in time t is dx/dt=sec²θdθ/dt. dθ/dt is the angular speed of the beacon=10π radians per minute, and dx/dt is the velocity of the beam along the shoreline. At x=2, dx/dt=(1+tan²θ)10π, and tanθ=2, so dx/dt=50π miles per minute=157 miles/min approx or about 9425 mph.