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.