A lake with $5$ x $108$ m$3$ water is fed by an upstream river having a flowrate of $20$ m$3/$s$.$ Effluent from a wastewater treatment plant is also discharged into the lake. The only source of phosphorus coming into the lake is effluent from the wastewater treatment plant, which carries $10$ mg$/$L phosphorus and is charged to the lake at $0\mathrm{.}4$ m$3/$s$.$ Phosphorus is removed from water via two pathways:

Phosphorus is taken up or removed by algae at a rate of $0\mathrm{.}2$ mg$/$m$3-$day.

Phosphorus forms insoluble precipitates with Ca$2+,$ Mg$2+$ and Fe$3+$ and the precipitates settle out of water at a rate $($vs$)$ of $10$ m$/$yr$.$ The surface area of the lake is $1$ x $108$ m$2.$

What is the steady state concentration of phosphorus in lake?