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

In summary, the problem involves a lighthouse located one mile from point P on a straight coastline. The light makes 4 revolutions per minute and we are trying to find the speed of the beam of light when it is 2 miles from point P. To solve this, we need to use the formula tan(\theta) = d/1 = d, where \theta represents the angle at which the light is pointing and d represents the distance up the shoreline. The distance from the lighthouse to the point 2 miles from P is Sqrt of 5, so we need to consider a circle with a radius of 5. Since the light makes 4 revolutions per minute, it will travel the circumference of this circle 4
  • #1
jmed
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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?


Homework Equations





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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  • #2
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 [itex]\theta= 0[/itex] when pointing at P, is [itex]tan(\theta)= 2/1= 2[/itex] or \(\displaystyle \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.\)
 
  • #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?
 
  • #4
what formula is going to be used to find the rate of the light beam?
 
  • #5
Any help?! I'm still unsure where to go with this?
 
  • #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...
 
  • #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. ...
 
  • #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
 
  • #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...
 
  • #10
any help? still lost
 

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

1. What is the concept of related rates in the context of a lighthouse?

Related rates refer to the relationship between different variables that are changing at the same time. In the context of a lighthouse, it involves analyzing how the height of the light beam and the distance between the lighthouse and a ship change as the ship moves closer or further away.

2. How is the distance between the lighthouse and the ship related to the height of the light beam?

The distance between the lighthouse and the ship is directly related to the height of the light beam. As the ship moves closer to the lighthouse, the distance decreases, and the light beam appears higher. Similarly, as the ship moves further away, the distance increases and the light beam appears lower.

3. What is the role of trigonometry in solving related rates problems involving a lighthouse?

Trigonometry is used to calculate the angles and sides of right triangles in related rates problems. In the case of a lighthouse, trigonometric functions such as sine, cosine, and tangent are used to determine the relationship between the height of the lighthouse, the distance between the lighthouse and the ship, and the angle of elevation of the light beam.

4. How can related rates be applied in real-life situations involving a lighthouse?

Related rates can be applied in real-life situations involving a lighthouse to calculate the rate of change of the light beam's height and the distance between the lighthouse and a moving ship. This information can be used to ensure the safety of the ship and to make necessary adjustments to the lighthouse's light beam.

5. What are some common mistakes to avoid when solving related rates problems involving a lighthouse?

One common mistake is to forget to consider all the given variables and their relationships. It is important to carefully read the problem and identify all the relevant information. Additionally, it is essential to use the correct units for measurements and to use the appropriate trigonometric functions based on the given information.

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