Pollutant $x$ is found in a lake $($with $50k{m}^3$ water$)$ besides a city. The lake receives water input from inflowing stream $($a river$),$ which contributes $Q\frac{m^3}{?}$ day water. However, this input doesn't change the lake water volume much since some of the lake water loses to groundwater and the air. $x$ is found in the inflowing stream $($the river$)$ to be $0\mathrm{.}01\frac{g}{m^3},$ steady everyday, but is not found in the air or groundwater. $x$ can be consumed by a kind of fish in the lake following a first$-$order decay:

change $of$ concentration $(c)$ due $to$ fish: $\frac{dc}{dt}=-0\mathrm{.}005c,in\frac{g}{m^3}\frac{?}{?}$ day

$($a$)$ Build a one$-$box model for the pollutant $x$ in this lake. Write down the continuity equation $)=($

$($b$)$ The concentration of $x$ in the lake is measured to be $0\mathrm{.}02\frac{g}{m^3},$ steady everyday. What is the average stream flow from the water $?)$

$($c$)$ At one day, a factory was open near the lake and released $x$ to the lake at the rate of $50k\frac{g}{?}$ day. Rewrite the continuity equation by adding this emission and the calculated value of $Q$ in $($b$).$

$($d$)$ With the initial concentration of $x,C_0=0\mathrm{.}02\frac{g}{m^3},$ what is the concentration after $30$ days? Please give the exact $($analytical$)$ solution.

$($e$)$ Solve $($d$)$ again numerically with the forward Euler scheme. Use a timestep of $10$ days.

$($f$)$ What's the error of this numerical solution in $($e$)?$ Is this method stable? Why?