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

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Homework Help Overview

The problem involves a lighthouse situated one mile offshore from a point on a straight coastline, with a light that rotates at a speed of 4 revolutions per minute. The question seeks to determine the speed of the light beam along the shoreline when it is 2 miles from the nearest point on the coast, point P.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the geometric relationship between the lighthouse, point P, and the beam of light, questioning the interpretation of distances and angles involved.
  • Some participants express confusion regarding the setup of the problem, particularly about the distance of the beam from the shoreline and the implications of the lighthouse's position.
  • There are attempts to relate the angle of the beam to the distances involved using trigonometric functions, with varying levels of clarity and correctness.
  • Questions arise about the appropriate formulas to use for determining the rate of the light beam's movement along the shoreline.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations and mathematical relationships. Some have provided insights into the geometry of the situation, while others are still seeking clarity on how to proceed with the calculations. There is no explicit consensus on the correct approach yet, but various lines of reasoning are being examined.

Contextual Notes

Participants are navigating through assumptions about the problem's setup, including the distances involved and the relationship between the lighthouse and the beam's movement. There is a noted lack of clarity regarding the definitions and relationships that govern the scenario.

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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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