Application of related rates

Question

Here is a question from a sheet my math teacher assigned me.

    A lighthouse is located on an island 4km away from the nearest point P on a straight shoreline. If its light makes 3 revolutions per minute, how fast is the beam of light moving along the shoreline when it is 750m from point P?  (the answer is supposed to be 78.05km/min.)  

For reference, here is the diagram I drew and used: https://i.stack.imgur.com/L2G0P.png . The point of the question is to find dl/dt.
My first step was to find h, simply using the Pythagorean theorem we can obtain an approximate value of 4069.7m.
Next, using sine law, I found the angle opposite 4000 (let's call this A) was 79.38degrees, and the angle opposite 750 (lets call this B) was 10.62degrees.
Due to the fact that the light makes 3rpm, I determined the rate of angle A to be 1080deg/min, and the rate of angle B to be -1080deg/min.
Finally, i used the sine law once again to determine: l/sinB = 4000/sinA.
(l)(sinA) = (4000)(sinB)
Differentiate implicitly:
(l)(cosA)(da/dt) + (dl/dt)(sinB) = 4000(cosB)(db/dt)
Solving for this equation didn't give me the correct answer. Where did I go wrong? Were some of my values incorrectly calculated or is the equation itself incorrect?
Thanks a lot,
Andrew

Answer 1

While your trigonometric calculations are correct, I am not sure whether your sine law and the rates that result from your implicit differentiation are correct.

I would use a simpler approach, just focusing on the rate of change of the angle swept by the beam from the lighthouse.

Examining the diagram below, note that the distance along the shoreline is simply $l=4000\tan(B)$, where $B$ is the angle swept by the beam from the lighthouse.

We are given that $\large\frac{dB}{dt}$ is $3$ revolutions per minute, corresponding to $6\pi$ radians per minute, resulting in $$l=4000\tan(B)\Rightarrow \frac{dl}{dt}=4000\sec^2(B)\left(\frac{dB}{dt}\right)=24000\pi\sec^2(B)$$ as the derivative of $\tan x$ is $\sec^2 x$

Substituting the value of $B$, which you found to be around $10.62^\circ$ into the above equation will result in the correct answer.