$(20$ Pts$)$ A drop control structure is need for a concrete lined trapezoidal canal with bottom width of $30$ feet, Manning's $n=0\mathrm{.}017,$ side slopes of $1\mathrm{.}5$ horizontal to $1$ vertical $(m=1\mathrm{.}5),$ and $S_0=0\mathrm{.}0007$ that conveys $450f\frac{t^3}{s}.$ The canal section will terminate in a free overfall over the edge of the drop structure.

a$.$ Determine the GVF profile by the direct step method using a y increment of $0\mathrm{.}1-$feet starting at the edge of drop. To avoid calculation instabilities, round your calculated starting depth up to the nearest $0\mathrm{.}1-$feet. Provide a plot or table with the $0\mathrm{.}1-$foot increment flow depths and corresponding distance $(x)$ from the drop.

b$.$ At what distance upstream from the drop does the flow reach a depth of $3\mathrm{.}0-$feet?