A lighthouse stands 500m off of a straight shore, the focused beam of its light

Question

A lighthouse stands 500m off of a straight shore, the focused beam of its light revolving four times each minute. P is the point on shore closest to the lighthouse and Q is a point on the shore 200 m from P. What is the speed of the beam along the shore when it strikes the point Q? Describe how the speed of the beam along the shore varies with the distance between P and Q. Neglect the height of the lighthouse.

Please show me the steps. It is very important to me to be able to fully understand how this problem is solved without shortcuts. Thank you very much for the help.

Explanation / Answer

so position of light on shore given by

x = y tan theta

at this point 200 = 500*tan(theta)

theta = arctan(2/5)=21.8 degrees

we know dtheta/dt = 2 pi/4

dx/dt = y sec(theta)^2 dtheta/dt

= 500*sec(21.8)^2 *2pi/4=911 m/min = 15.18 m/s

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