$($a$)$ A trapezoidal canal has to be designed to be of the most economical type. What is the basic geometric relationship which has to be ensured in the design? No proofs are required but a sketch should be given. $($b$)$ In a water resources development project, an irrigation canal has to be designed as a trapezoidal section with side slopes $1$:$1\mathrm{.}5($vertical: horizontal$).$ The bed slope is $1$ in $2000$ and the design discharge is $15$ m$3/$s$.$ If the canal section is unlined, Mannings n $=0\mathrm{.}025$ and if lined n $=0\mathrm{.}012,$ while the section is to be of the most economical type. Given these conditions; i$.$ Design the cross sections of the lined and unlined canals ii$.$ What is the percentage increase in volume of excavation and wetted area of the unlined canal, when compared to the lined canal?