Question $7(17$ marks$)$

A trapezoidal channel with side slopes of $3$ horizontal to $1\mathrm{.}67$ vertical has to be designed to convey $15\frac{m^3}{s}$ at a velocity of $1\mathrm{.}8\frac{m}{s},$ so that the amount of concrete lining for the bed and sides is minimum. Calculate the following:

$7\mathrm{.}1$ The wetted perimeter and

$7\mathrm{.}2$ The slope of the bed if Manning's $N=0\mathrm{.}014$ in the formula $C=\frac{1}{N}x{m}^{\frac{1}{6}}.$