Q $5-6$: The Manning equation can be written for a rectangular open channel as

$Q=\frac{\sqrt[\phantom{2}]{S}(BH{)}^{\frac{5}{3}}}{n(B+2H{)}^{\frac{2}{3}}}$

where $Q=$ flow $[\frac{m^3}{s}],S=$ slope $(\frac{m}{m}),B=$ width $(m),$ and $n=$ Manning roughness coefficient. Develop a fixed$-$point iteration scheme to solve this equation for $H$ given $Q=5,S=0\mathrm{.}0002,B=20,$ and $n=0\mathrm{.}03.$

$($a$)$ Articulate the engineering vs mathematical problem. $(3)$

$($b$)$ Prepare this function to apply Fixed Point Method $(3)$

$($c$)$ Write code to find a solution that $(14)$

$($i$)$ uses vectors and for loop I

$($ii$)$ plots the convergence over $20$ iterations with initial guess $H=1$

$($iii$)$ prints the final answer