Thread: Few pratice problems that I'm having trouble with.

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?


I don't get how to maximize in this equation.
A circle of radius r, you carve out a slice from the middle that makes angle 0<X<2pi, and has perimeter P. Maximize the area of the slice.

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·sec²θ·(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 .sec²θ = 10/9


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