A diversion gate backs up flow in a $5-$ft wide rectangular irrigation channel $($S$0=0\mathrm{.}001,$ n$=0\mathrm{.}015).$ The discharge in the channel is $50$ cfs$.$ If the depth of flow at the gate is $3\mathrm{.}5$ ft$,$ determine the channel and flow classification $($e$.$g$.,$ M$-2,$ S$-1,$ etc.$)$ upstream of the gate and explain your supporting logic. Then determine the water surface profile by finding the depth of flow at distances of $100,300,$ and $600$ ft upstream of the gate.