Problem $3(40$ points$)$:

Water discharge in a trapezoidal channel with a bottom width of $3m$ and side slopes $)$ of $3$ is $4\mathrm{.}3\frac{m^3}{s}.$ Given a bed slope of $0\mathrm{.}0008$ and a Manning roughness coefficient of $0\mathrm{.}016,$ determine the normal depth. Additionally, considering that the side slopes remain unchanged $),$ how can we design a more economical cross section for this channel based on the best hydraulic cross$-$section criteria?