Question

In: Math

A room has volume 100m3, and initially contains normal air. Every minute, 2m3 of gas is...

A room has volume 100m³, and initially contains normal air. Every minute, 2m³ of gas is pumped in, which is 1.9m³ air, and 0.1m³ xenon. Every minute, 2m³ of well-mixed gas is pumped out. What volume of xenon will be in the room after 1 hour?

Expert Solution

Let y(t) be the total volume(in ) of xenon in the room at t minutes.

Gas is pumped into the room at the rate 2 /min which is 1.9 air and 0.1 xenon. So, the rate at which xenon is pumped into the room is 0.1 /min.

The well-mixed gas is pumped out at the rate 2 /min.

The concentration of xenon in the air at time t minutes

So, the rate at which xenon is pumped out of the room is

Hence, the differential equation governing the system is

Since this is a separable differential equation, it can be solved as follows.

where C is the constant of integration.

At time t = 0, the room contains normal air. So, at t = 0, the total volume of xenon in the room is 0 i.e. y(0) = 0. Substituting this initial condition in our equation, we get

Now, the particular solution for the system is

The volume of the xenon in the room after 1 hour i.e. at t = 60 minutes is

milcah answered 2 years ago

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