Related rates ( revolving light)

Question

A revolving ligt located 10km from a straight shoreline turns at a constant angular speed of 3 rad/ min. How fast is the spot of the light moving along the shore when the beam makes an angle of pi/3 radians with the shoreline ?

Answer 1

Hi Alex,
 
Let me first try to draw a picture of this problem:
 
 
                                                        O  beacon
                                                        /|
                                                       / |
                                                      /  |
                                                  r  /θ | L
                                                    /    |
                                                   /     |
___________________________ /___ |______________  shore line

                                                     x
 
Now, L is the constant distance of the beacon from the straight shore. r is the path length of the current beam of light to the shore. Let θ be the instantaneous angle in radians that the light path makes with the vertical to the shore. Also, let x be the horizontal leg of this right triangle.

 

Let's write tanθ = x/L

 

Let's take the time derivative of each side of this last equation.

 

d/dt (tanθ) = d/dt (x/L)

 

For the left-hand side, d/dt (tanθ) = (dθ/dt)sec²θ, using the chain rule, understanding that θ is a time-dependent function.

 

For the right-hand side, we realize that L is a constant, so it can be pulled outside the derivative operator, and we can write: d/dt (x/L) = (1/L) dx/dt = v/L where v is the velocity that the light beam moves along the shore.

 

We now have

 

 

(dθ/dt)sec²θ = v/L

 

dθ/dt is the constant angular velocity, which we normally denote by ω. So, we can now write

 

ω sec²θ = v/L

 

Solving for the velocity, we get

 

v = ωL sec²θ

 

We know that

ω = 3 rad/min

L = 10 km

θ = pi/3 rad = 60 deg at the instant of interest.

 

So, the speed that the light beam moves along the shore at this instant is

 

v = (3 rad/min)(10 km)sec²(60 deg) = (3 rad/min)(10 km)(2²) = (3 rad/min)(10 km)(4) = 120 km/min

 

This is what I get.