Question

1. Consider the following figure below that shows two brine tanks r gal/min fresh water Tank 1 anks 2 containing V and V gallons of brine respectively. Fresh water flows into tank 1, while mixed brine flows from tank 1 into tank 2. Let x,() denote the amount (in pounds) of salt in tank i at time for i-1,2. If each flow rate is r gallons per minute, then a simple account of salt concentration yields the first-order system where If 50,玲= 25, r-10(ga// min) , and the initial amounts ofsalt in the two brine tanks, in pounds, are X,(0) = 15,划0) = 0, a) Find the amount of salt in each tank at time t2 0. b) Find the maximum amount of salt ever in tank 2.

Answer 1

Th $\begin{pmatrix} x_1'\\x_2' \end{pmatrix}=\begin{pmatrix} -1/5 &0 \\1/5 &-2/5 \end{pmatrix}\begin{pmatrix} x_1\\x_2 \end{pmatrix}$ e amount of salt concentrations yields the first order system

   $x_1'=-k_1x_1=-\frac{10}{50}x_1=-\frac{1}{5}x_1$                       -> (1)

   $x_2' =k_1x_1-k_2x_2= \frac{1}{5}x_1-\frac{2}{5}x_2$         -> (2)

The matrix form of the system is

$\begin{pmatrix} x_1'\\x_2' \end{pmatrix}=\begin{pmatrix} -1/5 &0 \\1/5 &-2/5 \end{pmatrix}\begin{pmatrix} x_1\\x_2 \end{pmatrix}$

Now to find the eigenvalues and eigen vectors:

Eigenvalues: $\lambda _1=-1/5, \lambda _2=-2/5$

Eigenvectors: $= \begin{pmatrix} 1\\1 \end{pmatrix}, \begin{pmatrix} 0\\1 \end{pmatrix}$

Thus the solution of the system of equation is given by

    $x(t)=C_1\begin{pmatrix} 1\\1 \end{pmatrix}e^{-1/5t}+C_2\begin{pmatrix} 0\\1 \end{pmatrix}e^{-2/5t}$