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Math Help - Related Rates 6

  1. #1
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    Related Rates 6

    A kite 116 ft above the ground moves horizontally at a speed of 11 ft/s. At what rate is the angle between the string and the horizontal decreasing when 232 ft of string have been let out?
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  2. #2
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    Hello, camherokid!

    This is a messy one . . . no wonder no one wants to take it on.


    A kite 116 ft above the ground moves horizontally at a speed of 11 ft/s.
    At what rate is the angle between the string and the horizontal decreasing
    when 232 ft of string have been let out?
    Code:
                      x
          * - - - - - - - - - - - *
          :                    *  :
          :                 *     :
          :           L  *        :
      116 :           *           : 116
          :        *              :
          :     *                 :
          :  * θ                  :
          * - - - - - - - - - - - *
                      x

    From the right triangle: .L .= .x + 116

    Differentiate with respect to time: .2L(dL/dt) .= .2x(dx/dt)

    . . Hence: .dL/dt .= .(x/L)(dx/dt)

    When L = 232, θ = π/6. .Hence: x = 116√3

    . . So: .dL/dt .= .[(116√3)/232](11) .= .(11√3)/2 . [1]


    We have: .cosθ .= .x/L . . Lcosθ .= .x

    Differentiate with respect to time: .cosθ(dL/dt) + L(-sinθ)(dθ/dt) .= .dx/dt . [2]


    When L = 232, θ = π/6. .Hence: sinθ = 1/2, cosθ = √3/2


    Substitute into [2]: .(√3/2)(11√3/2) + 116(-1/2)(dθ/dt) .= .11

    And we have: .dθ/dt .= .-11/464 . .-0.0237 radians/sec

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