Posted by deel on Wednesday, August 8, 2012 at 9:35pm.
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  MathMate, Wednesday, August 8, 2012 at 10:23pm
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*(20000y)/20000
Separate variables and integrate:
∫dy/(20000y) = (600/20000)∫ dt
log(20000y) = 600t/20000 + C'
log(20000y) = 600t/20000 + C
Raise to power of e:
20000y = e^(600t/20000+C)
y=20000(1e^(600t/20000+C)
At t=0, y=0
=>
0=(1e^(0+C))
=> C=0
Therefore:
y=20000(1e^(600t/20000))
at y=0.9*20000=18000,
=>
e^(600t/20000)=0.1
t=(20000/600)*log(0.1)
=77 minutes (approx.)

math  deel, Thursday, August 9, 2012 at 11:10am
thanks for your help

math :)  MathMate, Thursday, August 9, 2012 at 11:40am
You're welcome!
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