A trapezoidal channel has side slope 2 vertical to 3 horizontal.It is discharging water at the rate of 30 cunnees with a bed slope I in 2000. Design the channel for its best form .use Manning's formula, taking \( \mathrm{N}=0.01 \)

To design a trapezoidal channel using Manning's formula, you'll first need to determine the flow area and hydraulic radius. Given the trapezoidal shape with a side slope of 2:3, the bottom width (\(b\)) and depth (\(h\)) can be expressed in terms of the same variables. The discharge \(Q\) and Manning's equation relate as follows: \[ Q = \frac{1}{n} A R^{2/3} S^{1/2} \] Using \( Q = 30 \, \text{cubic units/second} \), \( n = 0.01 \), and a bed slope \( S = \frac{1}{2000} \), you can iteratively find the dimensions of the channel. Assuming \( h \) is the depth, the side slopes make the top width \( T = b + 2 * (h * (3/2)) = b + 3h \). The area becomes \( A = b * h + \frac{3}{2}h^2 \). You'll need to derive the hydraulic radius \( R \) by calculating \( R = \frac{A}{P} \), where \( P = b + 2h\sqrt{(3^2 + 2^2)} \). Balance \( Q \) with your findings, continuously refining \( h \) and \( b \) to optimize the channel design. To get the best results, don’t forget to validate your calculations with an instrument or software, as quick approximations while drafting designs can lead to mistakes. It’s always good to check your work! For further reading, check out Manning's equations and guides on open channel flow to enhance your understanding of the principles and applications. “Open Channel Hydraulics” by Robert J. Houghtalen is an excellent resource, providing practical examples and deeper insights into channel design!