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SCALC$95\mathrm{.}4\mathrm{.}030.$MI$.$

When gas expands in a cylinder with radius $r,$ the pressure $P$ at any given time is a function of the volume $V$:$P=P(V).$ The force exerted by the gas on the piston $($see the figure$)$ is the product of the pressure and the area: $F=r^{2}p.$

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The work done by the gas when the volume expands from volume $V_1$ to volume $V_2$ is as follows.

$w={\displaystyle \underset{V_1}{\overset{V_2}{}}}PdV$

In a steam engine the pressure and volume of steam satisfy the equation $P{V}^{1\mathrm{.}4}=k,$ where $k$ is a constant. $($This is true for adiabatic expansion, that is$,$ expansion in which there is no heat transfer between the cylinder and its surroundings.$)$ Calculate the work done by the engine during a cycle when the steam starts at a pressure of $160l\frac{b}{i}{n}^2$ and a volume of $100i{n}^3$ and expands to a volume of $900i{n}^3.($Round your answer to two decimal places.$)$

$w=$ ft $-$lb

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