Chapter 5, Problem 16P

Water is flowing in a trapezoidal channel at a rate of $Q=20{\text{m}}^3\text{/s}$. The critical depth $y$ for such a channel must satisfy the equation

$0=1-\frac{Q^2}{g{A}_c^3}B$

where $g=9.81{\text{ m/s}}^2,A_c=$ the cross-sectional area $\left({\text{m}}^2\right)$, and $B=$ the width of the channel at the surface $\left(\text{m}\right)$. For this case, the width and the cross-sectional area can be related to depth y by

$B=3+y\text{ and }{A}_c=3y+\frac{y^2}{2}$

Solve for the critical depth using (a) the graphical method, (b) bisection, and (c) false position. For (b) and (c) use initial guesses of $x_l=0.5\text{ and }{x}_u=2.5$, and iterate until the approximate error falls below 1% or the number of iterations exceeds 10. Discuss your results.