Page:Calculus Made Easy.pdf/184

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

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

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

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

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

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