I am having trouble working out why this differential equation models the situation given in the question. The question is that
The air of a room which has a volume of $~200~$m$^3$ contains $~0.15\%~$ CO$_2$. A fan delivers air containing $~0.04\%~$ at a rate of $~20~$m$^3$/min.
Apparently the answer is
$$ \frac{DQ}{Dt} = 0.008 - \frac{Q}{200}\cdot 20 $$
where $Q$ is the amount of CO2.
I get why the inflow rate is equal to $~20*0.0004~$ but I can't figure out why there is an output rate as the volume of the room is constant and no air is leaving. Any explanation would be appreciated.
I believe it's implicit that air is leaving at the same rate that air is coming into the room. This will maintain the amount of air, and the air pressure, at the default level (e.g., at or near sea level value) as, otherwise, the pressure will just keep increasing. This, of course, is not usually realistic unless there's some special mechanism in place to only add air without letting any of it leave.
Update: I believe using the verb "delivers" is somewhat misleading, as it implies just having air come into the room. Replacing "delivers" with something like "exchanges", so "A fan delivers air ..." becomes "A fan exchanges air ...", is likely a more accurate way to describe the action.