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Homework Help: Related rates and lighthouse

  1. Feb 23, 2010 #1
    1. The problem statement, all variables and given/known data

    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?


    2. Relevant equations



    3. 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.
     
  2. jcsd
  3. Feb 23, 2010 #2

    HallsofIvy

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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 lenght 1 and "opposite side" of length 2. At that point the angle the light is pointing, from [itex]\theta= 0[/itex] when pointing at P, is [itex]tan(\theta)= 2/1= 2[/itex] or [math]\theta= tan^{-1}(2)[/itex].

    In fact, at any time, the angle, [itex]\theta[/itex], and distance, d, up the shore are related by [itex]tan(\theta)= d/1= d[/itex]
    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.
     
  4. Feb 23, 2010 #3
    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?
     
  5. Feb 23, 2010 #4
    what formula is going to be used to find the rate of the light beam?
     
  6. Feb 23, 2010 #5
    Any help???! I'm still unsure where to go with this?
     
  7. Feb 23, 2010 #6
    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...
     
  8. Feb 24, 2010 #7
    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. ......
     
  9. Feb 24, 2010 #8
    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
     
  10. Feb 24, 2010 #9
    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...
     
  11. Feb 25, 2010 #10
    any help??? still lost
     
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