A concrete rectangular box culvert has a width of $1\mathrm{.}0$ m $,$ height of $0\mathrm{.}5$ m and a length of $30$ m with square$-$edged entrance conditions and free jet outlet flow.

Using the following data:

Manning's $\backslash (\backslash$mathrm$\{$n$\}=0\mathrm{.}013\backslash )$; and $\backslash (\backslash$mathrm$\{$C$\}\_\{\backslash$mathrm$\{$d$\}\}=0\mathrm{.}62\backslash )$

$1.$ Establish the stage $($headwater level$)-$discharge relationship for the culvert with a Slope $\backslash (=1\backslash )$ in $1000$ for a range of head water level for investigation: $\backslash (0-3\backslash$mathrm$\{$~m$\}\backslash ).$ Neglect the velocity of approach.

$2.$ Establish the stage $($headwater level$)-$discharge relationship for the culvert with a Slope $\backslash (=1\backslash )$ in $100$ for a range of head water level for investigation: $\backslash (0-3\backslash$mathrm$\{$~m$\}\backslash ).$ Neglect the velocity of approach.

$3.$ Plot the Performance Curve $($Stages vs Discharge$)$ for both Slope conditions.

$4.$ Analyze the two $(2)$ graphs $($rising and falling stages$).$

$5.$ Discuss the optimal suited application$($s$)$ for each slope condition.