$(25$ points$)$

The natural input to a lake has a relatively low concentration of phosphorous $($ P $),0\mathrm{.}2mg-\frac{P}{L}.$

However, a wastewater treatment plant discharges treated sewage containing $10mg-\frac{P}{L}.$ The

natural flow rate is ${10}^7\frac{m^3}{?}$ year, the wastewater flow rate is ${10}^6\frac{m^3}{y}$ear, and the lake's volume

is ${10}^8\frac{m^3}{?},$ and the surface area is ${10}^7{m}^2.$ Dust particles from nearby cities contain $P,$ and they are

deposited to the lake at the rate per unit area of $1mg-\frac{P}{yr-m^2}.$ The current situation has been in

effect for a time long enough to reach steady state. What is the $P$ concentration in the lake now?

Hint: Dust deposit does not follow conventional $n^{th}$ order kinetics. Take a closer look at the unit

of the reaction constant and think about the unit of the whole reaction term.