# math

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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 -

• math :) -

You're welcome!