Hello, mathman66!

A lighthouse is located 3km away from the nearest point P on a straight shoreline

and its light makes four revolutions per minute.

How fast is the beam of light moving along the shoreline when it is 1km from P? Code:

L
*
| *
| θ *
3 | *
| *
| *
* - - - - - - - - *
P x B

The lighthouse is at L.

Its beam shine at point B on the shore, x km from point P.

Let θ = angle BLP.

The light makes 4 revolutions per minute.

Each revolution is 2π radians.

Hence, dθ/dt = 8π radians/minute.

We have: .tanθ .= .x/3 . → . x .= .3·tanθ

Differentiate with respect to time: .dx/dt .= .3·sec˛θ·(dθ/dt) .**[1]**

When x = 1, we have a right triangle with angle θ.

The opposite side is 1, the adjacent side is 3.

. . Then the hypotenuse is √10.

Hence: .secθ = √10/3 .and .sec˛θ = 10/9

Substitute into [1]: .dx/dt .= .3(10/9)(8π) .= .80π/3 km/min