Question $1$: Open Channel Flow $-$ Optimal Flow Cross$-$Sections

$($a$)$ Given a rectangular open channel with a width $B,$ and a water depth $d,$ present an expression for the wetted perimeter $P.$

$[2]$

$($b$)$ When the wetted perimeter is at a minimum, show that $B=2D.$

$[4]$

$($c$)$ For a trapezoidal channel with bottom width $B,$ depth of water $d$ and side slopes $1(V)$ : $b(H),$ develop an expression for the wetted perimeter P in terms of $B,d,$ and $b.$

$[4]$

$($d$)$ For a minimum value of wetted perimeter $($i$.$e$.,d\frac{P}{d}d=0),$ express $B$ in terms of $d$ and $b.$

$[5]$

$($e$)$ Water flows along a rectangular channel at a depth of $1\mathrm{.}3$ m when the discharge is $8\mathrm{.}7\frac{m^3}{s}.$ There is a broad crested weir in the channel at a point where the flow discharge is required to be measured. The channel width $($B$)$ is $5\mathrm{.}5$ m $,$ and this is the same as the width of the weir $($b$).$ Ignoring energy losses, what is the minimum height $(y)$ of the broad crested weir if it is to function with critical depth on the crest?

$[10]$