A rectangular channel has a width of $5$ m$,$ carriers a discharge of $24$ m$3/$s with a depth of $3\mathrm{.}2$ m in Section $1.$ At a certain distance downstream $($Section $2),$ flow experiences a smooth and gradual contraction to a width of $3\mathrm{.}5$ m and a bed raise of $0\mathrm{.}5$ m$.$

Note: The energy loss is negligible.

a$)$ Calculate the flow depth in Section $2$ without considering the bridge piers $($y$2)[10].$

b$)$ Calculate the flow depth in Section $2$ with considering the minimum channel width $($yc$)[10]$

c$)$ Calculate the maximum diameter of the two bridge piers in Section $2$ such that no backwater occurs for this discharge $[20]?$