E3 Questions on Gradually varied flow  including integration of the backwater curve.

E3.1
A Rectangular channel is 3.0m wide, has a 0.01 slope, discharge of $5.3m^{3}/s$, and n=0.011. Find $y_{n}$ and $y_{c}$. If the actual depth of flow is 1.7m, what type of profile exists?
(Answer: $y_{n}$ = 0.4m, $y_{c}$ = 0.683m, S1 Curve) 
E3.2
A rectangular channel with a bottom width of $4.0m$ and a bottom slope of $0.0008$ has a discharge of $1.50m^{3}/s$. In a gradually varied flow in this channel, the depth at a certain location is found to be $0.30m$ assuming $n=0.016$, determining the type of GVF profile, the critical depth, and the normal depth.
(Answer: M2, $y_{c}=0.24m$, $y_{n}=0.43m$) 
E3.3
The figure below shows a backwater curve in a long rectangular channel. Determine using a numerical integration method, the profile for the following high flow conditions: $Q=10m^{3}/s$, $b=3m$, $n=0.022$, and a bed slope of $0.001$. Take the depth just upstream of the dam as the control point equal to $5m$. At what distance is the water level not affected by the dam? perform your integration using a) 2steps and b) 10steps
(Answer: $y_{n}=2.437m$.
Using $y_{0}$ (Euler) 2step $x=3490m$, 10step $x=4724m$.
Using $y_{1/2}$ 2step $x=4480m$, 10step $x=5906m$.
Using a forthorder Runge Kutta method gave 2step $x=3893m$, 10step $x=5461m$) 
E3.4
A trapezoidal, concretelined, channel has a constant bed slope of $0.0015$, a bed width of $3m$, and side slopes of $1:1$. A control gate increased the depth immediately upstream to $4m$. When the discharge is $19m^{3}/s$ compute the water surface profile upstream and identify the distance when the water depth is $1.8m$. ($n=0.017$)
(Answer: Using $y_{0}$ (Euler) 2step $x=1540m$, 10step $x=1695m$.
Using $y_{1/2}$ 2step $x=1670m$, 10step $x=1809m$.
Using mean GVF function: 2step $x=2729m$, 10step $x=1933m$.
Using a forthorder Runge Kutta method gave 2step $x=2023m$, 10step $x=1850m$) 
E3.5
Using the figure below, determine the profile for the channel conditions using a step length of $\Delta x=100m$. $Q=600m^{3}/s$, $n=0.04$, the bed slope of the rectangular channel is $S_{o}=0.002$ and has a width of $B=50m$. The sill height of the weir is $2.5m$ and the water depth over the weir is $4m$. Compare results from each method.