Q$3.($a$)$ A weir across an alluvial river has a horizontal floor of length $60$ m and retains $6$ m of water under full pond condition. If the downstream sheet pile is driven to a depth of $6$ m below the average bed level, calculate the exit gradient. Further, assuming porosity of $30$ percent and relative density of soil particles as $2\mathrm{.}70,$ estimate the critical exit gradient and the factor of safety of the system with respect to the exit gradient.

$(4\mathrm{.}5)$