Question

The volume

V
of a fixed amount of a gas varies directly as the temperature
T
and inversely as the pressure
P
. Suppose that
=V42cm3
when
=T84
kelvin and
=P8kgcm2
. Find the temperature when
=V74cm3
and
=P10kgcm2
.

Answer 1

`\bf \begin{array}{cccccclllll} \textit{something}&&\textit{varies directly to}&&\textit{something else}\\ \quad \\ \textit{something}&=&{{ \textit{some value}}}&\cdot &\textit{something else}\\ \quad \\ y&=&{{ k}}&\cdot&x && y={{ k }}x \end{array}\\ \quad \\`

and also


`\bf \begin{array}{llllll} \textit{something}&&\textit{varies inversely to}&\textit{something else}\\ \quad \\ \textit{something}&=&\cfrac{{{\textit{some value}}}}{}&\cfrac{}{\textit{something else}}\\ \quad \\ y&=&\cfrac{{{\textit{k}}}}{}&\cfrac{}{x} &&y=\cfrac{{{ k}}}{x} \end{array}`

now, we know that V varies directly to T and inversely to P simultaneously
thus
`\bf V=T\cdot \cfrac{k}{P}`

so
`\bf V=T\cdot \cfrac{k}{P}\qquad \begin{cases} V=42\\ T=84\\ P=8 \end{cases}\implies 42=\cfrac{84k}{8}\implies 4=k \\\\\\ V=\cfrac{4T}{P}\qquad now\quad \begin{cases} V=74\\ P=10 \end{cases}\implies 74=\cfrac{4T}{10}\implies 185=T`