Chapter 7.4, Problem 34E

In an unfortunate accident involving an Austrian truck, 100 kg of a highly toxic substance are spilled into Lake Sils, in the Swiss Engadine Valley. The river Inn carries the pollutant down to Lake Silvaplana and later to Lake St. Moritz.

This sorry state, t weeks after the accident, can be described by the vector
$\stackrel{\to }{x}(t)=\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\\ x_3(t)\end{array}\right]\begin{array}{c}\text{pollutant}\text{in}\text{Lake}\text{Sils}\\ \text{pollutant}\text{in}\text{Lake}\text{Silvaplana}\\ \text{pollutant}\text{in}\text{Lake}\text{St}\text{.}\text{Morits}\end{array}\}(\text{in}\text{kg}).$
Suppose that
$\stackrel{\to }{x}(t+1)=\left[\begin{array}{ccc}0.7& 0& 0\\ 0.1& 0.6& 0\\ 0& 0.2& 0.8\end{array}\right]\stackrel{\to }{x}(t)$
a. Explain the significance of the entries of the transformation matrix in practical terms.
b. Find closed formulas for the amount of pollutant in each of the three lakes t weeks after the accident.
Graph the three functions against time (on the same axes). When does the pollution in Lake Silvaplana reach a maximum?