Page:Calculus Made Easy.pdf/52

 are $$1570$$ cub. in. and $$2260.8$$ cub. in. respectively, and $$2260.8-1570=690.8$$.

Also, if

$r=h$,$\frac{dV}{dr}=2\pi r^2=400$ and $r=h=\sqrt{400}{2\pi}=7.98$ in.

(5) The reading $$\theta$$ of a Féry’s Radiation pyrometer is related to the Centigrade temperature $$t$$ of the observed body by the relation

$\frac {\theta}{\theta_1}=\left(\frac {t}{t_1}\right)^4$,

where $${\theta_1}$$ is the reading corresponding to a known temperature $$t_1$$ of the observed body.

Compare the sensitiveness of the pyrometer at temperatures $$800^\circ$$C., $$1000^\circ$$C., $$1200^\circ$$C., given that it read $$25$$ when the temperature was $$1000^\circ$$C.

The sensitiveness is the rate of variation of the reading with the temperature, that is $$\frac{d\theta}{dt}$$. The formula may be written $\theta=\frac{\theta_1}{t_1^4}t^4=\frac{25t^4}{1000^4} $, and we have $\frac{d\theta}{dt}=\frac{100t^3}{1000^4}=\frac{t^3}{10,000,000,000}$. When $$t=800$$, $$1000$$ and $$1200$$, we get $$\frac {d\theta}{dt}=0.0512$$, $$0.1$$ and $$0.1728$$ respectively.

The sensitiveness is approximately doubled from $$800^\circ$$ to $$1000^\circ$$, and becomes three-quarters as great again up to $$1200^\circ$$.