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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

.

Verified Answer :

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

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

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