# Math Help - Changing Atmosphere

1. ## Changing Atmosphere

I came up with my own problem.
There is nothing complicated.

Suppose a room (air tight) is filled with $100\%$ oxygen.
In the room there is a machine which takes in $5$ liters/minute of air and gives up pure nitrogen.

Find a way to model the amount of oxygen at any given time.
(Of course use obvious assumption necessary).

2. Nice problem indeed, thank you. I guess we assume that nitrogen and oxygen mix immediately in the whole volume.

The whole volume is $V$, the proportion of oxygen at time $t$ (in minutes) is $x(t)$, so $x(0)=1$ and probably $x(t)\to_{t\to\infty} 0$.

During an infinitesimal time interval $\Delta t$, the initial volume of oxygen $Vx(t)$ becomes $Vx(t+\Delta t)=Vx(t)-5\Delta t x(t)$ (since $5\Delta t$ is the volume that was taken away, among which a proportion $x(t)$ was composed of oxygen). So:
$\frac{x(t+\Delta t)-x(t)}{\Delta t}=-\frac{5}{V}x(t)$.

Stated differently, $\frac{dx}{dt}=-\frac{5}{V}x(t)$, so that $x(t)=e^{-\frac{5}{V}t}$. The proportion hence decreases exponentially fast.

Laurent.

3. Here is how I would do it.
Let $y(t)$ be a function which represents amount of oxygen at time $t$ in minutes.
Notice that $y'(t)$ is the rate of change.
Thus, $y'(t) = \text{ rate out } - \text{ rate in }$.
The rate in is amount of oxygen coming in per minute is $5\cdot \tfrac{y}{V}$.
The rate out is $0$.
Thus, $y'(t) = - \tfrac{5}{V}y$ with $y(0) = V$.