The emission rate of dioxin from a paper mill to a river is $25m\frac{g}{s}.$ The contaminant undergoes degradation with a first$-$order rate constant of $0\mathrm{.}6d^{-1}.$ The mean flow rate of the river is $5\frac{m^3}{s}$ and the mean water velocity is $0\mathrm{.}5\frac{m}{s}.$ Contaminant transport in the river can be adequately approximated by a one$-$dimensional advective model.

a$.$ Determine the concentration corresponding to the mean flow rate and mean water velocity at a downstream distance of $30km(2$ pts$,$ hint: find equation from lec $8$ slides in the Module of Canvas$).$

b$.$ If the standard deviation of the emission rate is $8m\frac{g}{s}$ and the standard deviation of the flow rate is $2\frac{m^3}{s},$ estimate the standard deviation of contaminant concentration at $30km(3pts,$ hints:

${}_c^2=(\frac{\text{d}\text{e}\text{l}\text{C}}{del{S}_0^{}}{)}^2{}_{S_0^{}}^2+(\frac{\text{d}\text{e}\text{l}\text{C}}{\text{d}\text{e}\text{l}\text{Q}}{)}^2{}_Q^2$

; express water velocity as function of flow rate and area of river; for differential: if $u$ and $v$ are both functions of $($:$x,\frac{d(uv)}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}\}$