What is the relationship between lighthouse location and beam rotation speed?

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SUMMARY

The discussion focuses on calculating the speed of a lighthouse beam along a shoreline when it is 2 miles from point P, given that the lighthouse is 1 mile offshore and the beam makes 4 revolutions per minute. The relationship between the angle of the beam and the distance along the shore is established using trigonometric functions, specifically tan(θ) = d/1, where d is the distance along the shore. The circumference of the circle traced by the beam is calculated, leading to the conclusion that the beam moves at a speed of 4 times the circumference of the circle with a radius of √5 miles.

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Homework Statement



A lighthouse is located off shore one mile from the nearest point P, on a straight coastline. The light makes 4 revolutions per minute. How fast is the beam of light moving along the shoreline when it is 2 miles from point P?


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The Attempt at a Solution



I am confused. Is this asking when the beam of light is basically rotated a half revolution making the beam 2 miles from the shoreline? Not sure how to set it up.
 
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The beam is NOT "two miles from the shore line". The beam crosses the shoreline at a point 2 miles up the coast from point P. Since the lighthouse itself is 1 mile from P, you have a right triangle with "near side" of length 1 and "opposite side" of length 2. At that point the angle the light is pointing, from \theta= 0 when pointing at P, is tan(\theta)= 2/1= 2 or [math]\theta= tan^{-1}(2)[/itex].

In fact, at any time, the angle, \theta, and distance, d, up the shore are related by tan(\theta)= d/1= d
I have no idea where you got the "half revolution". If you are taking "pointing at P" to be the starting position, a half revolution would have the light pointing directly away from the coast line.
 
so the hypotnuse is the unknown (d). the distance up the shoreline is 2. the distance from P to the light house is 1. How is the tan of theta d/1? wouldn't it be 2/d?
 
what formula is going to be used to find the rate of the light beam?
 
Any help?! I'm still unsure where to go with this?
 
i mean I can't give you the desired related rates method, but if the light is making 4 revoultions per minute, that means that it travels around a circle 4 times in a minute. so if you consider a circle with a radius of 2 miles (since the point is 2 miles away), the light will travel across that point 4 times in a minute (starting from that point) . So find the circumference of that circle, the light will travel the circumference of the circle 4 times in a minute...
 
ok, so the circumference is 2 pi and that divided by 4 is pi/2. So I'm still confused on how to find how fast the beam is moving when 2 miles from point P.? The distance from the lighthouse to the point 2 miles from P is Sqrt of 5. ...
 
ok i misread a little bit of the problem, but I got it now, you have the point p is sqrt(5) miles away from the light house, so consider a circle from light house of radius 5, the light will make a circle 4 times in a minute, meaning it will travel the length of the circle of circumference sqrt(5) 4 times in one minute. so 4 times the circumference divided by a minute
 
A lighthouse is located off shore one mile from the nearest point P, on a straight coastline. The light makes 4 revolutions per minute. How fast is the beam of light moving along the shoreline when it is 2 miles from point P?

lighthouse is one mile from point P...trying to find how fast beam of light is moving along shoreline when 2 miles from point P...
 
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any help? still lost
 

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