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Math Help - Few pratice problems that I'm having trouble with.

  1. #1
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    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.
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  2. #2
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    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

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