Problem $2$: A river receives a wastewater discharge that has an ultimate BOD of $\backslash (50\backslash$mathrm$\{$mg$\}/\backslash$mathrm$\{$L$\}\backslash ).$ The river has an ultimate DO of $\backslash (9\backslash$mathrm$\{$mg$\}/\backslash$mathrm$\{$L$\}\backslash ),$ a degradation rate constant of $0\mathrm{.}2$ per day, a velocity of $\backslash (0\mathrm{.}4\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ),$ a depth of $3$ meters, and a temperature of $\backslash (25^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$

$1.$ Calculate the critical time at which the minimum DO occurs.

$2.$ Calculate the minimum $\backslash ($ D O $\backslash )$ in the river.

$3.$ Calculate the O'Connor$-$Dobbins re$-$aeration rate constant.