Chapter 9.1, Problem 15E

Justify the “Rule of 69”: If a quantity grows at a constant instantaneous rate of k%, then its doubling time is about $69/k$ . Example: In 2008 the population of Madagascar was about 20 million, growing at an annual rate of about 3%, with a doubling time of about $69/3=23$ years.

Consider the system
$\frac{d\stackrel{\to }{x}}{dt}=\left[\begin{array}{cc}{\lambda }_1& 0\\ 0& {\lambda }_2\end{array}\right]\stackrel{\to }{x}$