Best hydraulic rectangular canal design

For the best hydraulic rectangular section, width is twice the depth: $b = 2y$, so $A = 2y^2$ and $R = y/2$.

Manning's equation

$$\begin{aligned} Q &= \frac{1}{n}AR^{2/3}S^{1/2} \\ 15 &= \frac{1}{0.02}(2y^2)\left(\frac{y}{2}\right)^{2/3}(0.001)^{1/2} \\ 15 &= 50 \times 2 \times 0.63 \times 0.03162\, y^{8/3} = 1.992\,y^{8/3} \\ y^{8/3} &= 7.529 \;\Rightarrow\; y = 7.529^{3/8} = 2.132\ \text{m} \end{aligned}$$

Section

$$\begin{aligned} b &= 2 \times 2.132 = 4.264\ \text{m} \\ A &= 2 \times 2.132^2 = 9.091\ \text{m}^2, \quad R = 1.066\ \text{m} \\ V &= 15/9.091 = 1.65\ \text{m/s} \end{aligned}$$

Check: $50 \times 9.091 \times 1.066^{2/3} \times 0.03162 = 15.0$ m³/s.

   |<---- b = 4.26 m ---->|
   |~~~~~~~~~~~~~~~~~~~~~~|
   |                      |  y = 2.13 m
   |______________________|

Answer: Width ≈ 4.26 m, depth ≈ 2.13 m (plus freeboard), velocity ≈ 1.65 m/s.