To calculate the bed width of a hydraulically efficient trapezoidal channel section, we need to understand the concept of hydraulic efficiency and the relationship between flow depth and channel dimensions.



Hydraulic efficiency refers to the ability of a channel to transport water with minimum energy losses and resistance. In a hydraulically efficient channel, the flow is smooth and uniform, resulting in the highest possible conveyance capacity for a given channel cross-section.



In this problem, we are given that the flow depth in the channel is 2 m. Uniform flow occurs when the water surface is parallel to the channel bed, and the flow depth remains constant along the channel length.



A trapezoidal channel is a common channel shape used in hydraulic engineering. It consists of a trapezoidal cross-section with a sloping side and a flat bottom. The channel is defined by its flow depth, bottom width, and side slope.



To calculate the bed width of the channel, we need to use the Manning's equation, which relates the flow rate, channel dimensions, and flow characteristics. The equation is as follows:

Q = (1.49/n) * A * R^(2/3) * S^(1/2)

Where:
Q = Flow rate (m^3/s)
n = Manning's roughness coefficient (dimensionless)
A = Cross-sectional area of flow (m^2)
R = Hydraulic radius (m)
S = Channel slope (m/m)

In a hydraulically efficient channel, the flow is uniform, and the cross-sectional area is maximized for a given flow depth. For a trapezoidal channel, the cross-sectional area is given by:

A = y * (b + z * y)

Where:
y = Flow depth (m)
b = Bottom width of the channel (m)
z = Side slope of the channel (dimensionless)

To maximize the cross-sectional area, we need to choose appropriate values for the bottom width (b) and side slope (z). In this case, the question states that the channel is hydraulically efficient, which means that the dimensions have been optimized for the given flow depth.



To find the bed width of the channel, we can rearrange the equation for the cross-sectional area and solve for b:

A = y * (b + z * y)
b = (A/y) - z * y

We know that the flow depth (y) is 2 m. However, we don't have any information about the cross-sectional area (A) or the side slope (z). Therefore, we cannot determine the exact value of the bed width (b).

However, the correct answer is given as between 2.29 m and 2.32 m. This range suggests that the bed width has been optimized to achieve hydraulic efficiency for a flow depth of 2 m. The specific value within this range will depend on the specific channel dimensions and