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## Problem

Suppose the volume $V$ and the pressure $P$ of a gas undergoing adiabatic compression is modelled by ${PV}^{1.2} = C$, where $C$ is constant. When the volume is 64 L, the pressure is 20 kPa.

If at a moment the pressure is decreasing at the rate of 1 kPa per second, find the instantaneous rate of change of the volume.

## Solution

What is known?

$\frac{dP}{dt} = -1 \: kPa/s$ at $P = 20 \: kPa$ and $V = 64 \: L$.

Derive the function with respect to time.

$\frac{d}{dt} \left(PV^{1.2} \right) = \frac{d}{dt} (C)$

Since $C$ is constant,

$\frac{dP}{dt} V^{1.2} + P(1.2) V^{0.2} \frac{dV}{dt} = 0$
$(-1){(64)}^{1.2} + (20)(1.2){(64)}^{0.2} \frac{dV}{dt} = 0$
$\frac{dV}{dt} = \frac{64}{24}$
$\frac{dV}{dt} = \frac{8}{3} \: L/s$
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