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 $$y=xa^{\frac{1}{x}}$$.