Assume a lake has a total volume of $1,000km3.$ Suppose the lake is fed by three rivers with flow rates, FaFb, and $F_C$ corresponding to $10,000m^{3}da{y}^{-1},60,000m^{3}da{y}^{-1}$ and $90,000m^3{d}_{ay}^{-1},$ respectively. The lake has one outlet. Given that the three rivers bring into the lake three inert pollutants $x,Y$ and $Z,$ as follows: $Y$ from River $A$ at a concentration of $C_{Ya}=10m\frac{g}{L}$;$x$ from River B at a concentration of $C_{xb}=2m\frac{g}{L}$; and $x$ and $Z$ from River $C$ at concentrations of $C_{xc}=8m\frac{g}{L}$ and $C_{Zc}=4m\frac{g}{L},$ respectively.

Additionally, two factories discharge effluents into the reservoir containing the three inert pollutants as follows;

Factory $1$ has a flow rate, F$1,$ equal to $800m^{3}da{y}^{-1},$ with pollutants $Y$ and $Z$ at concentrations, $C_{YI}$ and $C_{Zl},$ equal to $24mg{L}^{-1}$ and $10mg{L}^{-1},$ respectively;

Factory $2$ has a flow rate, F$2,$ equal to $500m^{3}da{y}^{-1},$ with pollutants $x,Y$ and $Z$ at concentrations, $C_{x2},C_{Y2}$ and $C_{Z2},$ equal to $7mg{L}^{-1},8m\frac{g}{L}$ and $9mg{L}^{-1},$ respectively;

a$)$ Calculate the pollutant concentration $x,Y$ and $Z$ in the lake. Make the system sketch and set the system boundaries and all the necessary assumptions for the calculation. $[20$ marks$]$