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Math Help - Air flow in a room. Differential equations

  1. #1
    Newbie leebatt's Avatar
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    Air flow in a room. Differential equations

    The air in a small 12 ft by 8 ft by 8 ft room is 3% carbon monoxide. Starting at t = 0, fresh air containing no carbon dioxide is blown into the room at a rate of 100 cubic feet per minute. If air in the room flows out through a vent at the same rate, when will be the air in the room be 0.01% carbon monoxide?


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  2. #2
    MHF Contributor
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    volume of room ... 12 \cdot 8^2 = 768 \, ft^3

    let r = percentage of room air changed in 1 minute ...

    r = \frac{100}{768} \cdot 100 \approx 13.02%

    let p = percentage volume of CO in the room at any time t

    \frac{dp}{dt} = -rp

    p = p_o e^{-rt}

    p = 3e^{-rt}

    .01 = 3e^{-rt}

    t = \frac{\ln{300}}{r} \approx 43.8 \, min
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  3. #3
    Eater of Worlds
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    Here's another way using the integrating factor thing. More roundabout than the skeets though.

    \frac{dy}{dt}=\text{rate in-rate out}

    The rate in of the air is \left(\frac{100 \;\ ft^{3}}{min}\right)

    \text{rate out}=\left(\frac{y(t)}{768} \frac{ft^{3}}{ft^{3}}\right)\left(\frac{100 \;\ ft^{3}}{min}\right)=\frac{25y}{192}

    The equation is then:

    \frac{dy}{dt}-\frac{25y}{192}=100

    The integrating factor is e^{\int \frac{25}{192}dt}=e^{\frac{25t}{192}}

    Use the IC to solve for C. The amount of air at t=0 is

    768(.97)=744.96.

    ye^{\frac{25t}{192}}=768e^{\frac{25t}{192}}+C

    Using y(0)=744.96, we find that C=-23.04

    We have \boxed{y=768-23.04e^{\frac{-25t}{192}}}

    .01% CO means 99.99% air. .9999(768)=767.9232

    767.9232=768-23.04e^{\frac{25t}{192}}

    Solving or t we find \boxed{t=43.805 \;\ min.}

    You can also do this in terms of the CO instead of air. Either way should give the same solution.
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