Question $1[$Non$-$Linear Algebraic equations$]$

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

channel must satisfy the equation:

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

where $g=9\mathrm{.}81\frac{m}{s^2},A{c}_c=$ the cross$-$sectional area $(m^2),$ and $B=$ the width of the channel at the

surface $(m).$ For this case, the width and the cross$-$sectional area can be related to depth $y$ by

$B=3+y,A_c=3y+\frac{y^2}{2}$

Solve for the critical depth using $($a$)$ the graphical method, $($b$)$ bisection, and $($c$)$ false position.

For $($b$)$ and $($c$),$ use initial guesses of $x_{\text{I}}=0\mathrm{.}5$ and $x_u=3,$ and iterate until the approximate error

falls below $1\%$ or the number of iterations exceeds $10.$ Discuss your results using matlab