A sluice gate is positioned in a rectangular channel with width $B=6m.$ It Figure $1.$ Definition sketch of the rectangular ahannel with sluice gate $($not to scale$).$discharges a flow rate of $Q=20\frac{m^3}{s}.$ You can neglect any friction losses. Figure $1$ below illustrates the sluice gate system. For this case, complete the following tasks:

a$)$ Calculate the downstream flow depths $y_2$ and $y_3$ and the corresponding Froude numbers $F{r}_2$ and $F{r}_3$ assuming that the depth $y_3$ is equal to the natural water level in the channel downstream of the structure.

b$)$ Sketch the sluice gate system and add the total head line as well as the water surface profile.

c$)$ Calculate the force on the sluice gate.

In a second case, the natural water level in the channel downstream of the sluice gate is increased by a tailwater control far downstream of the sluice gate. For this second case, complete the following tasks:

d$)$ Sketch the flows along the sluice gate system and include any relevant features and describe the key changes between the case shown in Figure $1$ and this second case.

e$)$ Which of the two flow cases is more efficient? Briefly explain your answer.