$(20$ pts$)$ A ${12}^{\text{'}}$ wide rectangular concrete canal flowing $4\mathrm{.}2$ feet deep and conveying $200f\frac{t^3}{sec}$ must cross under a roadway without increasing the upstream flow depth. Assuming a smooth transition $($no losses$)$ a$)$ Would an $8\text{'}$ wide box culvert $($rectangular channel$)$ work for the crossing? If so$,$ what would the flow depth in the box culvert be$?$

b$)$ Would a $7\text{'}$ wide box culvert $($rectangular channel$)$ work? Explain and show calculations.