$(25$ pts$)$ Two streams flow into a $400-$gallon tank. The first stream is $15\mathrm{.}0wt\%$ ethanol and $85\mathrm{.}0\%$ hexane $($the mixture density, ${}_1,$ is $0\mathrm{.}65\frac{g}{c}{m}^3)$ and the second is $75\mathrm{.}0wt\%$ ethanol, $25\mathrm{.}0\%$ hexane $0$:$75\frac{g}{c}{m}^3).$ After the tank has been filled, which takes $22$min, an analysis of its contents determines that the mixture is $40\mathrm{.}0wt\%$ ethanol, $60\mathrm{.}0\%$ hexane. You wish to estimate the density of the final mixture and the mass and volumetric flow rates of the two feed streams.

$($a$)$ Draw and label a flowchart of the mixing process and do the degree$-$of$-$freedom analysis.

$($b$)$ Perform the calculations and state what you assumed.