$($b$)$ Consider the stirred$-$tank blending process. The nominal steady$-$state conditions are $w_1=600k\frac{g}{m}in,w_2=2k\frac{g}{m}in,x_1=0\mathrm{.}05,$ and $x_2=1($for pure solute$).$

The liquid volume and density are constant: $V=2m^3$ and $p=900k\frac{g}{m^3},$ respectively.

$($i$)$ Calculate the nominal exit concentration, $\stackrel{}{x}.$

$($ii$)$ Derive an expression for the response, $x(t),$ to a sudden change in $x_1$ from $0\mathrm{.}05$ to $0\mathrm{.}075$ that occurs at time, $t=0.$

Assume that the process is initially at the nominal steady state.