Nitrate $(N{O}_3)$ is a common groundwater contaminant in rural areas. Sources of $N{O}_3$ include

fertilizers, septic systems, and manure. Nitrate is regulated in drinking water because it

causes methemoglobinemia, or "blue baby" syndrome. The water standard for nitrate is $10$

$m\frac{g}{L}.$ Consider a small lake which receives input from a river at rate $Q_R=0\mathrm{.}1\frac{m^3}{s}$ and from a

groundwater seepage rate $Q_G=0\mathrm{.}005\frac{m^3}{s}.$ The river inflow is free of nitrate, but the

groundwater contains nitrate percolated from septic tanks around the lake. The mean nitrate

concentration in the groundwater is $40m\frac{g}{L}.$ The lake volume is ${10}^5{m}^3.$

a$)$ Assuming the nitrate concentration has reached a steady state in the lake, what is the

nitrate concentration in the lake outflow?

b$)$ If the groundwater source could be instantly removed, approximately how long would it

take for the nitrate concentration in the lake to drop to zero?