Q$2($a$).$ Show analytically, using the mass balance equation for reservoir inflow$-$outflow and reservoir storage, why does the peak of the outflow discharge$-$hydrograph intersect the falling limb of the inflow hydrograph? $(8)$

Q$2($b$).$ For a radial gate in closed position, as shown in the figure, compute the horizontal and vertical forces on the trunnion axis of the gate if the water level in the reservoir is $1\mathrm{.}5m$ above the crest of the spillway. What is the reaction force on the spillway crest? Consider $1m$ length normal to the paper. $(3+3+2)$

Q$2($c$).$ What are the geometric profiles of the spillway surface to the right and left of its crest $($mathematical representation of the two curves$)?(4)$