What is the speed of the lighthouse beam along the shoreline?

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SUMMARY

The discussion revolves around calculating the speed of a lighthouse beam along a shoreline, specifically when the beam is 1km from point P, located 3km from the lighthouse. The light makes four revolutions per minute, leading to a rate of change of the angle (dθ/dt) of 24π radians per minute. The user derived the relationship using the tangent function and ultimately calculated the speed of the beam along the shoreline (dx/dt) to be 80π, which raised concerns about its plausibility.

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Feodalherren
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Homework Statement


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


Homework Equations





The Attempt at a Solution



I drew a picture and got a triangle out of it. I labeled the center angle θ and a straight line from P that's 1km long as X.

Tan θ = x/3


Want dx/dt when dθ/dt = 24∏

correct?
The reasoning was that
C = 2∏r

r=3

C=6∏ but 4 revolutions/min gives me 24∏.

I get dx/dt = 80∏ as the solution, that seems way too fast.
 
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Feodalherren said:

Homework Statement


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


Homework Equations





The Attempt at a Solution



I drew a picture and got a triangle out of it. I labeled the center angle θ and a straight line from P that's 1km long as X.

Tan θ = x/3


Want dx/dt when dθ/dt = 24∏

correct?
The reasoning was that
C = 2∏r

r=3

C=6∏ but 4 revolutions/min gives me 24∏.

I get dx/dt = 80∏ as the solution, that seems way too fast.

4 revolutions per minute gives 2∏ radians per 15 seconds. What does that give you for ω in units of radians per second?
 
Hmm I'm not following... Could you tell me how you arrived at that?
 

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