009A Sample Final 2, Problem 5

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A lighthouse is located on a small island 3km away from the nearest point    on a straight shoreline and its light makes 4 revolutions per minute. How fast is the beam of light moving along the shoreline on a point 1km away from  

When we see a problem talking about rates, it is usually a related rates problem.
Thus, we treat everything as a function of time, or  
We can usually find an equation relating one unknown to another, and then use implicit differentiation.
Since the problem usually gives us one rate, and from the given info we can usually find the values of

variables at our particular moment in time, we can solve the equation for the remaining rate.


Step 1:  
We can begin this physical word problem by drawing a picture.
009A SF2 5GP.png
In the picture, we can consider the distance from the point    to the spot the light hits the shore to be the variable  
By drawing a right triangle with the beam as its hypotenuse, we can see that our variable

   is related to the angle    by the equation

This gives us a relation between the two variables.
Step 2:  
Now, we use implicit differentiation to find
Rearranging, we have
Again, everything is a function of time.
Step 3:  
We want to know the rate that the beam is moving along the shore when

we are one km away from the point  

This tells us that  
The problem also tells us that the lighthouse beam is revolving at 4 revolutions

per minute.

However,    is measured in radians, and there are    radians in a revolution.
Thus, we know
Finally, we require secant. Since we know  
we can solve the triangle to get that the length of the hypotenuse is
This means that
Step 4:  
Now, we can plug in all these values to find


Final Answer:  

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