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 $1200^{\circ }$.