Trapezoidal channel design with Chezy's formula

Only $Q$, $S$, side slope and $C$ are given, so one more condition is needed. The channel is designed as the most economical trapezoidal section (standard assumption): $R = y/2$ and half the top width = one sloping side.

Proportions (side slope $n = 1$)

$$\begin{aligned} \frac{b + 2ny}{2} &= y\sqrt{1+n^2} \;\Rightarrow\; b = 2y(\sqrt{2} - 1) = 0.8284y \\ A &= (b + ny)y = (0.8284 + 1)y^2 = 1.8284y^2 \\ R &= y/2 \end{aligned}$$

Chezy's equation

$$\begin{aligned} Q &= AC\sqrt{RS} \\ 5 &= 1.8284y^2 \times 56 \times \sqrt{\frac{y}{2} \times \frac{1}{1200}} \\ 5 &= 1.8284 \times 56 \times 0.020412\, y^{2.5} = 2.0901\, y^{2.5} \\ y^{2.5} &= 2.392 \;\Rightarrow\; y = 1.418\ \text{m} \end{aligned}$$

Dimensions

$$\begin{aligned} b &= 0.8284 \times 1.418 = 1.174\ \text{m} \\ A &= 1.8284 \times 1.418^2 = 3.674\ \text{m}^2, \quad R = 0.709\ \text{m} \\ V &= 5/3.674 = 1.361\ \text{m/s} \end{aligned}$$

Check: $3.674 \times 56 \times \sqrt{0.709/1200} = 5.00$ m³/s.

  |<------- T = 4.01 m ------->|
  \                            /
   \  y = 1.42 m         1:1  /
    \________________________/
         b = 1.17 m

Answer: Bottom width ≈ 1.17 m, depth of flow ≈ 1.42 m, side slopes 1:1 (top width ≈ 4.01 m). In practice a freeboard of about 0.3–0.5 m is added.