Calculus help!?

Question

So I am struggling with this question (the 4 rpm thing is throwing me off):

"A lighthouse is located on a small island 3 km away from the nearest point P on a straigh shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P?"

So far I have drawn out the right triangle and made the angle between the beam of light and the perpindicular theta. Made the verticle line (perpindicular to the shore) 3 km, and the shore "x". I think I should be solving for dx/dt? Anyway, I've used "tangent theta = x/3" to relate everything. Made that into x = 3 tangent theta, took the derivative, and solved for dx/dt (after calculating theta). I came up with 38.317.... but I don't know the units and I'm pretty sure it is not correct. Any help would really be appreciated.

Answer 1

Let
L – point where lighthouse is located
A – point on the shoreline 1 km from point P
ω – angular speed of the light beam
ω = 8 π /min
Distance LA is √10 (Pythagoras!).
The speed of light beam at point A perpendicular to the beam itself is
LA * ω = √10 * 8 π (km/min)
The angle between this speed and the shoreline is the same as angle Cosinus of this angle is
cos ( The speed of the light beam along the shoreline is
LA * ω / cos ( -