(9) The damping on a telephone line can be ascertained from the relation ${\displaystyle i=i_{0}\epsilon ^{-\beta l}}$, where ${\displaystyle i}$ is the strength, after ${\displaystyle t}$ seconds, of a telephonic current of initial strength ${\displaystyle i_{0}}$; ${\displaystyle l}$ is the length of the line in kilometres, and ${\displaystyle \beta }$ is a constant. For the Franco-English submarine cable laid in ${\displaystyle 1910}$, ${\displaystyle \beta =0.0114}$. Find the damping at the end of the cable (${\displaystyle 40}$ kilometres), and the length along which ${\displaystyle i}$ is still ${\displaystyle 8\%}$ of the original current (limiting value of very good audition).

(10) The pressure ${\displaystyle p}$ of the atmosphere at an altitude ${\displaystyle h}$ kilometres is given by ${\displaystyle p=p_{0}\epsilon ^{-kh}}$; ${\displaystyle p_{0}}$ being the pressure at sea-level (${\displaystyle 760}$ millimetres).

The pressures at ${\displaystyle 10}$, ${\displaystyle 20}$ and ${\displaystyle 50}$ kilometres being ${\displaystyle 199.2}$, ${\displaystyle 42.2}$, ${\displaystyle 0.32}$ respectively, find ${\displaystyle k}$ in each case. Using the mean value of ${\displaystyle k}$, find the percentage error in each case.

(11) Find the minimum or maximum of ${\displaystyle y=x^{x}}$.

(12) Find the minimum or maximum of ${\displaystyle y=x^{\frac {1}{x}}}$.

(13) Find the minimum or maximum of ${\displaystyle y=xa^{\frac {1}{x}}}$.