a$)$ Prove that the most efficient section or the best hydraulic section for a trapezoidal channel is determined when the top water width $($ T $)$ is twice the side length $($w$)$ and is half of a hexagon with a side slope of $60$ and z is $\frac{1}{\sqrt[\phantom{2}]{3}}$ on both sides.

b$)$ Storm water from a housing area is discharging $10\mathrm{.}8\frac{m^3}{s}$ and is designed to flow in a trapezoidal section with Manning's n is $0\mathrm{.}018,$ and the bed slope of $0\mathrm{.}0015.$ Find the water depth of the best hydraulic section.