Q$1(5$ points$)$: An open channel with a bottom width $5m$ and Manning's $n0\mathrm{.}013m^{-\frac{1}{3}}$ discharges $10\frac{m^3}{s}$ water to a community located in a rural area, the channel's slope is $0\mathrm{.}005.$ Find the normal depth of the channel using the Newton Raphson method. To start iterations, use an initial depth of $0\mathrm{.}05m,$ error of $0\mathrm{.}01,$ and $N$ of $100$ for:

a$)$ A rectangular channel

$Q=\frac{1}{n}\frac{b{h}^{\frac{5}{3}}}{(1+2\frac{h}{b}{)}^{\frac{2}{3}}}{S}^{\frac{1}{2}}$

b$)$ A trapezoidal channel

$Q=\frac{1}{n}{h}^{\frac{2}{3}}(\frac{1+2\frac{h}{b}}{1+2\sqrt[\phantom{2}]{5}\frac{h}{b}}{)}^{\frac{2}{3}}{S}^{\frac{1}{2}}hb(1+2\frac{h}{b})$