$(20$ pts$)$ You have been tasked with designing a new section of trapezoidal canal with the following characteristics:

Slightly Sinuous

Side Slopes of $2$ Horizontal to $1$ Vertical $(m=2)$

Design Flow $=50ft\frac{3}{sec}$

Manning's $n=0\mathrm{.}027$

Channel Slope $=0\mathrm{.}001f\frac{t}{f}t$

Canal Material is Sandy Clay $($Cohesive$)$ with a $0\mathrm{.}7$ Void Ratio

a$.$ Using the tractive force method, determine a design bottom width and flow depth

b$.$ Check that the Froude number is within typical design parameters

c$.$ Check the channel sides and bed for erosion

d$.$ Determine the freeboard using the US Bureau of Reclamation $(1959)$ methodology