Problem $3.$ The approximate profile of a stream channel is given in the graph below. The depth is modeled using the equation

$d(w)=\frac{1}{8}(w^4-20w^3+141w^2-406w)+51$

$(******)$ At pre$-$flood stage, the water depth is $20ft($the green line$),$ and the edge of the stream spans from $w=\mathrm{.}8ft$ to $w=9ft.$ Calculate the cubic feet per second if the water is moving at $9$ feet per second. Repeat this process for "typical" early summer flows $(10ft$ depth marked in purple$)$ when the river spans from $w=1\mathrm{.}3ft$ to $w=8\mathrm{.}5ft,$ and the water moving at $5$ feet per second.

$(**)$ You can see from the profile that the stream depth is not uniform. Calculate the average depth at "typical" early summer flow. Draw a picture showing the simplified stream profile. Use your simplified stream depth model to create an equation that gives the cubic feet per second which relies only on stream speed $v.$

What are the limitations of your model?