A rectangular, concrete lined channel is to be constructed to carry floodwater. The slope of the ground surface is $1$ in $500.$ The design discharge is $10\frac{m^3}{s}.$

a$.$ Calculate the proportions of the rectangular channel that will minimise excavation and result in the optimum hydraulic section. $($Answer: $D=1\mathrm{.}444m$ and $B=2^{***}Dm).$

b$.$ If the cross$-$section area of flow is kept the same, but for safety reasons the depth of flow in the channel is limited to $1\mathrm{.}00m,$ what will be the discharge? $($Answer: $9\mathrm{.}47\frac{m^3}{s}).$