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Math Help - Stuck on a Related Rates w/ Trig Problem!!

  1. #1
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    Stuck on a Related Rates w/ Trig Problem!!

    I am completely lost on this one:

    Tutor.com | Session Transcript - Math - Calculus, 11/15/2009 7:46PM
    In the engine below a 7 inch connecting rod is fastened to a crank of radius 3 inches. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when theta= pi/3. You will need to use the law of cosines.




    I don't really even get how to draw this as a triangle - PLEASE HELP!!
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  2. #2
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    Well, use the triangle you've got in the middle of your pic, with theta the angle between the horizontal and the crank. The crank looks realistically of length 3 relative to the connecting rod of length 7, and this rod is the side opposite the angle theta. Call the length of the horizontal side x. Then the cosine rule

    \cos A = \frac{b^2 + c^2 - a^2}{2bc}

    gives us

    \cos \theta = \frac{9 + x^2 - 49}{2*3x} = \frac{x^2 - 40}{6x} = \frac{x}{6} - \frac{20}{3x}

    Related rates nearly always depend on the chain rule, so you might want to try filling up this pattern...



    ... where straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case time), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

    So express one variable as a function of the other...



    Then differentiate with respect to the inner function, and the inner function with respect to t. Then you can sub in 1/2 for the dashed balloon expression and for x you can sub the very nice value that you get from solving

    \cos(\frac{\pi}{3}) = \frac{x^2 - 40}{6x}

    (for x). Hope that helps.

    Spoiler:



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