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Math Help - another question:)

  1. #1
    Newbie
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    Nov 2008
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    another question:)

    As television cable is pulled from a large spool to be strung from the telephone poles along a street,it unwinds from the spool in layers of constant radius.If the truck pulling the cable moves at a steady 6ft/sec (a touch over 4 mph),use the equation s=rθ to find how fast (radians per second )the spool is turning when the layer of radius 1.2ft is being unwound.
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, turkeyy!

    As a cable is pulled from a large spool, it unwinds in layers of constant radius.

    If the cable is pulled at a steady 6 ft/sec, use the equation s\:=\:r\theta
    to find how fast (radians/sec) the spool is turning when the radius is 1.2 ft.
    Code:
                  * * *
              *           *
            *               *
           *                 *
    
          *         O         *
          *         *         *
          *         | *       *
                    | θ *
           *    1.2 |     *  *
            *       |       o Q
              *     |     *
                  * o * → → → → → →
                    P     6 ft/s

    The cable is pulled horizontally at 6 ft/sec.
    Then point P is moving around the circle at the same rate.

    We have: . r = 1.2

    \text{Let }\,s \:=\:\text{arc(PQ)}\text{, then: }\,\frac{ds}{dt} \,=\,6


    From s \:=\:r\theta, we have: . s \:=\:1.2\,\theta

    Differentiate with respect to time: . \frac{ds}{dt} \:=\:1.2\,\frac{d\theta}{dt} \quad\Rightarrow\quad \frac{d\theta}{dt} \:=\:\frac{1}{1.2}\,\frac{ds}{dt}

    Since \frac{ds}{dt} = 6, we have: . \frac{d\theta}{dt} \:=\:\frac{1}{1.2}(6) \:=\:5\text{ rad/sec}


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