$11.(40$ pts$.)$ Consider an office whose HVAC system provides $5$ air changes per hour to a $\backslash (40\backslash$mathrm$\{$~m$\}^\{3\}\backslash )$ conference room. Herein, $\backslash (80\backslash \%\backslash )$ of the supply air is recirculated, and the other $\backslash (20\backslash \%\backslash )$ is outdoor air ventilation. A filter removes particles from the supply air stream with a $\backslash (75\backslash \%\backslash )$ efficiency. At $9$ am $,$ an employee lights a scented candle in this room that emits $6$ mg of particles every hour. This conference room hosts a meeting from noon to $1$ pm $.($Assume: $\backslash (\backslash$beta$=0\mathrm{.}5\backslash$mathrm$\{$~h$\}^\{-1\}\backslash )$ for particles in this room; occupants breathe at a rate of $\backslash (0\mathrm{.}6\backslash$mathrm$\{$~m$\}^\{3\}/\backslash$mathrm$\{$h$\}\backslash )$ during the meeting. Neglect outdoor particles.$)$

a$)$ When will the particle concentration reach $\backslash (99\backslash \%\backslash )$ of its steady state concentration?

b$)$ What is the intake fraction $($IF$)$ of particles for an occupant during this meeting?

c$)$ What would the IF be if the meeting were instead from $9$am$-10$am$?$