$(20$ pts$)$ A flash drum is a one$-$stage distillation column and is used quite often in industry. Its streams are F$,$ V$,$ and L for Feed, Vapor, and Liquid, respectively. In the diagram pictured below, the feed flow rate is $1,800l{b}_m$min and the vapor stream flow rate is $5001\frac{b_m}{m}in.$ The feed has a mass fraction composition, $z_1,$ of $0\mathrm{.}4.$

a$.$ If the mass fraction of component one in the vapor stream, $y_1,$ is $0\mathrm{.}8,$ what is the total flow rate of the liquid stream and its composition $($i$.$e$.,x_1)?$

b$.$ If the density of the liquid stream is $48l\frac{b_m}{f}{t}^3,$ what is the volumetric flow rate in $f\frac{t^3}{m}in?$