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.

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.)

thanks for your help

You're welcome!

Te amo :3

To find the amount of fresh air inside the room at any time, we need to consider the rate of fresh air flowing in and the rate of air being exhausted.

Let's define the function y(t) as the amount of fresh air inside the room at time t. Since we are given that y(0) = 0, it means there is no fresh air in the room initially.

To determine the rate of change of fresh air inside the room, we can use the concept of differential equations. The rate of change of fresh air, dy/dt, can be expressed as the inflow rate minus the outflow rate:

dy/dt = Inflow rate - Outflow rate.

We know that the inflow rate is 600 ft^3 per minute, and the outflow rate is also 600 ft^3 per minute. Therefore, we can write the differential equation as:

dy/dt = 600 - 600.

Simplifying, we have:

dy/dt = 0.

This differential equation tells us that the rate of change of fresh air inside the room is constant, i.e., there is no change in the amount of fresh air over time.

Since the rate of change is 0, it means that the amount of fresh air inside the room remains constant at 0 ft^3. In other words, the room will never have any fresh air based on the given parameters.

As a result, we cannot determine the time required for 90% of the air in the room to be fresh because there will never be any fresh air in the room.