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Thread: Word problem including depth, min / max..

  1. #1
    Nov 2007

    Word problem including depth, min / max..

    One end of a piece of elastic is attached to a point at the top of a door frame and the other end hangs freely. A small ball is attached to the free end of the elastic. When the ball is hanging freely it is pulled down a small distance and then release, so that the ball oscillates up and down on the elastic. The dept d cm of the ball from the top of the door frame after t seconds is given by:

    d = 100 + 10 cos 500 t


    (a) the greatest and least depths of the ball
    Solution: Found 'em to be 110 and 90, respectively, when cos part is -1 or 1.

    (b) the time at which the ball first reaches it highest position.
    Solution: That would be at 90, which occurs when the cos equation first gets -1, i.e. when t = 180 / 500 = 0.36

    (c) the time taken for a complete oscillation
    Solution: This would be when t = 360 / 500 = 0.72

    (d) the proportion of the time during a complete oscillation for wihch the depth of the ball is less than 99cms.

    I need help with (d) part.
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  2. #2
    Senior Member Twig's Avatar
    Mar 2008



    You need the propotion of the time.
    Canīt one solve for $\displaystyle t_{1} $ when depth is 99cm and
    $\displaystyle t_{2} $, that is, the second time the depth is 99cm.

    Then time spent "over" the 99cm mark is $\displaystyle t_{2} - t_{1} $ .

    $\displaystyle \mbox{Proportion } = \frac{t_{2} - t_{1}}{0.72} $
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