Chapter 7, Problem 30P

An environmental engineer wishes to blend a single mixture of insecticide spray solution of volume $V=500\text{L}$ with a specified concentration $C=0.2$ from two spray solutions of concentration $c_1=0.15$ and $c_2=0.25$. The required volumes or the two spray solutions ${\nu }_1$ and ${\nu }_2$ can be determined from a system of equations describing conditions for volume and concentration, respectively as

${\nu }_1+{\nu }_2=V$ $(7.136)$ $c_1{\nu }_1+c_2{\nu }_2=CV$ . $(7.137)$ (a) Knowing that $V=500\text{L}$, $c_1=0.15$, $c_2=0.25$, and $C=0.2$, rewrite the system of equations $(7.136)$ and $(7.137)$ in terms of ${\nu }_1$ and ${\nu }_2$.

(b) Find ${\nu }_1$ and ${\nu }_2$ using the substitution method.

(c) Write the system of equations obtained in part (a) in the matrix form $Ax=b$, where $x=\left[\frac{{\nu }_1}{{\nu }_2}\right]$.

(d) Find ${\nu }_1$ and ${\nu }_2$ using the matrix algebra method. Perform all computations by hand and show all steps.

(e) Find ${\nu }_1$ and ${\nu }_2$ using the Cramer’s rule.