Consider a dam where there is a continuous upstream flow to it $($Figure $3).$ There is a sluice gate of width $w$ at its side $($not shown on the graph$).$ The engineer raises the sluice gate such that it creates an opening of a at the bottom, and the area of the opening is comparable with the dam's opening. It was measured that the depth of the water in the dam is y$1,$ whereas the depth of the water downstream is y$2.$ Find the volume flow rate. $($Hint: you can assume the flow is in equilibrium, i$.$e$.,$ the flow rate upstream and downstream is identical$)(8$ points$)$