QUESTION $1[25$ MARKS$]$

$($a$)$ A triangular channel of apex $90$ and a rectangular channel as shown in Figure $1$ have the same material and bed slope. If the rectangular channel has a depth equal to the width of the channel, and if cross$-$sectional areas in both channels are the same, calculate the ratio of discharges in the rectangular and triangular channels, respectively.

$[15$ Marks$]$

$($b$)$ Derived the simplied final equations for the channel depth, cross$-$sectional area, wetted perimeter, and hydraulics radius in terms of the channel width, B for the most economic trapezoidal channel section. Show the details derivations beginning from:

$y=\frac{Bsin}{2(1-cos)}$

$[10$ Marks$]$