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Math Help - Figuring out an equation relating 3 variables

  1. #1
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    Figuring out an equation relating 3 variables

    I have a practical problem that I need to set up an equation for, any help on how to do this would be really appreciated.

    This is the scenario- I have a tube that I am injecting fluid into at the top. After I have injected the fluid it moves down the tube at a constant speed. This fluid makes the part of the tube it is currently passing through visible to a measuring device I have. When the fluid reaches the point where I have my observation device setup, the device turns on and starts moving. The device moves down the tube a constant speed that is less than that of the fluid. The device stays on for a pre-determined amount of time.

    V1 = speed of liquid (cm/sec)
    V2= speed of observation device (cm/sec)
    A = Time spend injecting the liquid (sec)
    C= Time interval of observation (sec)
    (V1*A) = length of liquid column as it moves down tube (cm)
    (V2*C) = length of the section of tube observed by the device (cm)
    The length of the tube can be assumed to be infinite for the purposes of this problem. The interval of interest can start at any point after I start injecting the liquid. t = 0 is where the observation device turns on.

    During the the time interval C, the only condition that has to be met is that my device always has to have some part of the liquid column in its view. The goal is to minimize the injection time A, so that as little liquid as possible is used. How can I set up an equation that relates these variables in a way which allows me to figure out the shortest allowed A, when V1, V2, and C are known.
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  2. #2
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    Re: Figuring out an equation relating 3 variables

    To get the smallest possible A, you want the trailing end of the liquid column to pass the device just as it turns off. Also note that the length of the column of liquid stays the same.

    At this point, I would usually draw a couple pictures - what it looks like at t=0 and what it looks like at t=C. In this time, as you said, the device moves V2*C. How much does the liquid move? So you should be able to figure out the length of the column, and from that the injection time A.

    - Hollywood
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