# Derivative/rate problem

#### daigo

##### Member
A lighthouse 1 km from a beach shore revolves at 10$$\pi$$ radians/minute. How fast is the light sweeping across the shore 2 km from the lighthouse?
I drew a diagram here to try and help:

I think it will have something to do with a triangle. So just to find the other side, I'll use the Pythagorean theorem and get $$\sqrt{3}$$. I already knew this because I recognized the 30-60-90 triangle, so I know the angles too, but I don't know if we need them yet.

What I don't understand is what the question is asking. I thought the light was going around at a constant speed of 10$$\pi$$ radians/minute? What speed do they want?

#### CaptainBlack

##### Well-known member
A lighthouse 1 km from a beach shore revolves at 10$$\pi$$ radians/minute. How fast is the light sweeping across the shore 2 km from the lighthouse?
1. Don't put the quetion in a quote, it makes it more difficult to construct a coherent reply.

2. $$v=\dot{\theta}r$$, $$\dot{\theta}$$ in units of radians per unit time, is probably what they are looking for, but there is really insufficient information in what you posted to be sure.

CB

#### HallsofIvy

##### Well-known member
MHB Math Helper
I drew a diagram here to try and help:

I think it will have something to do with a triangle. So just to find the other side, I'll use the Pythagorean theorem and get $$\sqrt{3}$$. I already knew this because I recognized the 30-60-90 triangle, so I know the angles too, but I don't know if we need them yet.

What I don't understand is what the question is asking. I thought the light was going around at a constant speed of 10$$\pi$$ radians/minute? What speed do they want?
You are asked how fast the light is moving along the shoreline- that would be a speed in km per min or km per hr. What you are given is how fast the light is rotating. Yes, at the moment shown, the section along the shortline that the light reaches is 2 times $$\sqrt{3}$$. But the light is moving so that right triangle is changing. If you call the angle theta, do you see that opposite leg of the triangle is given by $tan(theta)$? What is the rate of change of that length relative to the rate of change of the angle.