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

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

Compare the rate at which ${\displaystyle P}$ 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 ${\displaystyle r}$;

(b) the area of a circle of radius ${\displaystyle r}$;

(c) the lateral area of a cone of slant dimension ${\displaystyle l}$;

(d) the volume of a cone of radius ${\displaystyle r}$ and height ${\displaystyle h}$;

(e) the area of a sphere of radius ${\displaystyle r}$;

(f) the volume of a sphere of radius ${\displaystyle r}$.

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