Problem $2$: In this task, we will compute the depth of a flow in an open channel $(y)$ for a given flowrate $($Q$)$ using Manning's equation given as:

$Q=\frac{1}{n}A{R}^{\frac{2}{3}}{S}_o^{\frac{1}{2}}$

Where n is the Manning's roughness coefficient, A is the wetted$-$area of the rectangular channel, $R$ is the hydraulic radius which is the ratio of wetted$-$area $($ A $)$ and wetted$-$perimeter $(P)$ of the channel and $S_0$ is the slope of the channel. For a rectangular channel with a width of $b$ and a flow depth of $y,$ area $($A$),$ wetted perimeter $($P$)$ and hydraulic radius $($R$)$ are given as follows:

$A=by$;$P=b+2y$;$R=\frac{A}{P}=\frac{by}{b+2y}$

Substituting the above parameters in the Q equation above, we have

$Q=\frac{1}{n}(by)(\frac{by}{b+2y}{)}^{\frac{2}{3}}{S}_o^{\frac{1}{2}}$

Find the design depth $(y)$ of a rectangular channel made with reinforced concrete $(n=0\mathrm{.}013)$ for a flowrate of $5\frac{m^3}{s}($Q$),$ width $($b$)$ of $5$ m and a slope $So=0\mathrm{.}001$