An industrial process discharges its effluent into a stream. It is desired to determine the effects of the waste discharge on the dissolved oxygen concentration of the stream. $k_1$ at $20C$ is $0\mathrm{.}23d^{-1},$ and $k_2$ at $20C$ is $0\mathrm{.}43d^{-1}.$ The characteristics of the stream and industrial wastewater are given below.

$\backslash$table$[[$Characteristics$,$Stream,Industrial wastewater$],[$Flow rate, $\frac{m^3}{s},6\mathrm{.}5,0\mathrm{.}5],[$Temperature$,C,19\mathrm{.}2,25\mathrm{.}0],[$Dissolved oxygen, $m\frac{g}{L},8\mathrm{.}2,0\mathrm{.}5],[BO{D}_5$ at $20C,3\mathrm{.}0,200\mathrm{.}0]]$

a$.$ Calculate the critical values and the distance from the point of discharge at which the critical values will occur. The stream velocity is $0\mathrm{.}2\frac{m}{s}.$

b$.$ Calculate the values and draw the dissolved oxygen sag curve for a $100km$ reach of the stream from the point of discharge.