A nearly pristine river with a saturated DO concentration of $11\mathrm{.}5$ mg$/$L receives a sewagedischarge from a wastewater treatment plant. Immediately upstream of the discharge, theriver flow is $2\mathrm{.}1$ m$3/$s$.$ The river temperature remains constant before and after receiving thedischarge. The upstream river water has an ultimate BOD of $0\mathrm{.}5$ mg$/$L$,$ and a dissolvedoxygen concentration of $8\mathrm{.}4$ mg$/$L$.$ The depth and width of the river are quite uniform at $1\mathrm{.}5$m and $30$ m$,$ respectively, throughout its length. The wastewater discharges at a flow of $0\mathrm{.}6$m$3/$s with an ultimate BOD of $45$ mg$/$L and a dissolved oxygen concentration of $1\mathrm{.}5$ mg$/$L$.$The sewage being discharged has a kd of $0\mathrm{.}45$ d$-1$ at $20\backslash$deg C$.$ The river and the wastewatertemperature are at $25\backslash$deg C$.$a$.$ Calculate the BOD remaining and dissolved oxygen concentration $($both in mg$/$L$)1\mathrm{.}0$km downstream of the point where the sewage enters the river.b$.$ At what distance $($in km$)$ is the dissolved oxygen at its minimum concentration?