$3.$ A point source Gaussian plume model for a power plant uses $50$ m as the effective stack height. You are concerned about pollutant levels at night, and the night is typically fully overcast with a wind speed of $\backslash (5\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$ Your concern in with ground$-$level pollution at two house locations, $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash ),$ which are $1\mathrm{.}2$ and $1\mathrm{.}4$ km directly downwind from the stack. Figure $7\mathrm{.}52$ in the Masters $\backslash$& Ela text would be helpful.

$($a$)$ At what distance will the maximum concentration of pollution occur under these conditions? Which location $($A or B$)$ would have the higher level of pollution?

$($b$)$ Suppose during the day the sky clears up$,$ and the atmosphere becomes moderately unstable, and the wind speed stays around $\backslash (5\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$ Will the location downwind at which the maximum concentration occurs move? If so$,$ will it move closer to the stack or further away from the stack? $($H remains $50$ m $).$

$($c$)$ Under the new conditions in part b$,$ which house would experience the most pollution?