Page:Calculus Made Easy.pdf/54

(10) The greatest external pressure $P$ which a tube can support without collapsing is given by

$P=\left({\frac {2E}{1-\sigma ^{2}}}\right){\frac {t^{3}}{D^{3}}}$,

where $E$ and $\sigma$ are constants, $t$ is the thickness of the tube and $D$ is its diameter. (This formula assumes that $4t$ is small compared to $D$.)

Compare the rate at which $P$ varies for a small change of thickness and for a small change of diameter taking place separately.

(11) Find, from first principles, the rate at which the following vary with respect to a change in radius:

(a) the circumference of a circle of radius $r$;

(b) the area of a circle of radius $r$;

(c) the lateral area of a cone of slant dimension $l$;

(d) the volume of a cone of radius $r$ and height $h$;

(e) the area of a sphere of radius $r$;

(f) the volume of a sphere of radius $r$.

(12) The length $L$ of an iron rod at the temperature $T$ being given by $L=l_{t}\left[1+0.000012(T-t)\right]$, where $l_{t}$ is the length at the temperature $t$, find the rate of variation of the diameter $D$ of an iron tyre suitable for being shrunk on a wheel, when the temperature $T$ varies.