Explaining the physical meaning of an eigenvalue in a real world problem Contextual Problem
A PhD student in Applied Mathematics is defending his dissertation and needs to make 10 gallon keg consisting of vodka and beer to placate his thesis committee.  Suppose that all committee members, being stubborn people, refuse to sign his dissertation paperwork until the next day.   Since all committee members will be driving home immediately after his defense, he wants to make sure that they all drive home safely.  To do so, he must ensure that his mixture doesn't contain too much alcohol in it!  
Therefore, his goal is to make a 10 gallon mixture of vodka and beer such that the total alcohol content of the mixture is only $12$ percent.  Suppose that beer has $8\%$ alcohol while vodka has $40\%$.  If $x$ is the volume of beer and $y$ is the volume of vodka needed, then clearly the system of equations is 
\begin{equation}
x+y=10 \\ 0.08 x +0.4 y = 0.12\times 10 
\end{equation}
My Question
The eigenvalues and eigenvectors of the corresponding matrix
\begin{equation}
\left[
\begin{array}{cc}
1 & 1\\
0.08 & 0.4
\end{array}
\right]
\end{equation} 
are
\begin{align}
\lambda_1\approx 1.1123 \\
\lambda_2\approx 0.2877 \\
v_1\approx\left[\begin{array}{c}
0.9938 \\ 0.1116
\end{array}
\right] \\
v_2\approx\left[\begin{array}{c}
-0.8145 \\ 0.5802
\end{array}
\right]
\end{align}
How do I interpret their physical meaning in the context of this particular problem?

First we need to interpret the transformation "physically". What this does is gets the amount of each type of alcohol as input, and spits out the total volume and alcohol volume as output.
I'd be surprised if this has a serious "physical interpretation", because the input and output are of different types. I suppose I'd say that the eigenvectors are the alcohol mixes where the ratio of beer to vodka is the same as the percentage of alcohol in the final mixture. Technically interprable, but probably no more than a curiosity.
