a$)$ Derive a mathematical formula for the maximum mixing depth $($MMD$)$ for a hot, dry air pollution source $($Tsource $>$ Tambient$)$ into a subadiabatic atmosphere based on Tsource, Tambient, $($dT$/$dz$)$adiabatic and $($dT$/$dz$)$ambient$.$

b$)$ Consider the emission of exhaust from a vehicle. The temperature of the exhaust is $50$ degrees $C$ and the exhaust behaves adiabatically. The ambient temperature is $25$ degrees $C$ and the ambient lapse rate is $-5$ degrees $\frac{C}{k}m.$ What is the expected MMD for the emissions.

c$)$ Do you think the emissions will rise to that predicted height? Why or why not?