Problem $1(15$ pts$)$

A ventilation system with air conditioning is used to cool down a classroom with $35$ persons. The room dimensions are $4^{**}{4}^{**}3m.$ Cold air enters the room at a rate $=q$ and leaves at the same rate. The heat generated by each person is assumed to be constant $=E_g.$ The room is assumed to be insulated so that no heat enters from outside. Temperature outside the room and initial room temperature $=T_2.$ Cold temperature enters the room at a temperature $=T_c.$

a$.$ Sketch the system and label it appropriately. $1pt$

b$.$ Derive and solve the relationship $($symbolic$)$ between room temperature and time: $T(t).7$ pts

c$.$ If the air condition suddenly shuts off $($no air enters or leaves the room$),$ Derive the relationship between room temperature and time: $T(t)\mathrm{.}4$ pts

d$.$ Estimate the required ventilation in $\frac{m^3}{s}$ needed to keep room temperature below $23C$ assuming Tinitial $=T$ ambient $=29C,T$ cold in $=21C,Ep=110W,p$ air $=1\mathrm{.}24k\frac{g}{m^3},C_p$ air $=1k\frac{J}{k}g.K$ below the allowable limit$.3$ pts