To solve for the critical depth $y$ of the trapezoidal channel, we are tasked with finding the root of the given equation using the bisection method. The equation for the critical depth $y$ can be written as:

$0 = 1 - \frac{Q^2}{gA_c^3B}$

Where:

• $Q = 20 \, \text{m}^3/\text{s}$ (flow rate),
• $g = 9.81 \, \text{m}/\text{s}^2$ (gravitational constant),
• $A_c = 3y + \frac{y^2}{2}$ (cross-sectional area),
• $B = 3 + y$ (surface width).

We are asked to perform the bisection method with the initial guesses $x_l = 0.5$ and $x_u = 2.5$, iterating until either the approximate error is less than 1%, or the number of iterations exceeds 10.

Steps to Follow:

1. Define the function: We need to express the equation in terms of $y$, where: $f(y) = 1 - \frac{Q^2}{gA_c^3B}$ with the given expressions for $A_c$ and $B$.

2. Bisection Method Setup:

   • Initial lower bound $x_l = 0.5$,
   • Initial upper bound $x_u = 2.5$,
   • Stopping criteria: relative error less than 1% or 10 iterations.

3. Iteration Process:

   • Compute the midpoint $x_r = \frac{x_l + x_u}{2}$,
   • Evaluate $f(x_r)$,
   • Depending on the sign of $f(x_r)$, adjust the bounds and calculate the approximate error $\varepsilon$.

Let me now proceed to solve this iteratively using Python to implement the bisection method.The bisection method converged to a solution after 10 iterations. The critical depth $y$ is approximately $0.6699 \, \text{m}$, and the approximate error after the final iteration was $0.29\%$, which is below the 1% tolerance threshold.

Here’s a summary of the results:

• Final critical depth: $y \approx 0.6699 \, \text{m}$
• Approximate error: $0.29\%$
• Number of iterations: 10

Would you like further details on the iteration process or have any specific questions?

Here are 5 related questions to explore further:

1. How does changing the initial guesses affect the convergence of the bisection method?
2. What would happen if the flow rate $Q$ were increased?
3. How does the bisection method compare to other root-finding methods like Newton's method?
4. Can the critical depth equation be solved symbolically for specific cases?
5. What are the physical implications of a critical depth in open-channel flow?

Tip: The bisection method is simple but requires many iterations for high precision. Faster convergence methods like Newton-Raphson can be more efficient but require good initial guesses.