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In a room of 20.000 ft^3 of air, 600 ft^3 of fresh air flows in per minute, and the mixture (made practically inform by circulating fans) is exhausted at a rate of 600 cubic feet per minute (cfm). What is the amount of fresh air inside the room y(t) at any time, if y(0)=0? After what time will 90% of air in the room be fresh.

  • math -

    I assume you are familiar with integration or differential equations.

    y(t)=amount of air (in c.f.) in room at time t.

    Then
    dy/dt
    =y'(t)
    =rate of increase of fresh air per unit time (min)
    =fresh air in - fresh air out per minute
    =600 - 600(y/20000)
    =600*(20000-y)/20000
    Separate variables and integrate:
    ∫dy/(20000-y) = (600/20000)∫ dt
    -log(20000-y) = 600t/20000 + C'
    log(20000-y) = -600t/20000 + C
    Raise to power of e:
    20000-y = e^(-600t/20000+C)
    y=20000(1-e^(-600t/20000+C)
    At t=0, y=0
    =>
    0=(1-e^(0+C))
    => C=0
    Therefore:
    y=20000(1-e^(-600t/20000))
    at y=0.9*20000=18000,
    =>
    e^(-600t/20000)=0.1

    t=-(20000/600)*log(0.1)
    =77 minutes (approx.)

  • math -

    thanks for your help

  • math :) -

    You're welcome!

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