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Problem 38 Medium Difficulty

When air expands adiabatically (without gaining or losing heat), its pressure $ P $ and volume $ V $ are related by the equation $ PV^{1.4} = C, $ where $ C $ is a constant. Suppose that at a certain instant the volume is $ 400 cm^3 $ and the pressure is $ 80 kPa $ and is decreasing at a rate of $ 10 kPa/min. $ At what rate is the volume increasing at this instant?

Answer

The volume is increasing at 35.7 $\mathrm{cm}^{3} / \min$

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Video Transcript

Okay, we know pressure and temperature fall in this equation is PV to the 1.4 is equivalent to see. Therefore, we know that we have to determine Devi over DT, right? We're turning the rate in this equation there for recon, right? Devi over. DT is t d p over DT, which we know is negative 10 times V, which is 400 over p which is 80 times 1.4, which I wrote appears the exponents, which is equivalent to 35.7 and the unit is centimeters Q put minute. So we know the volume has to be increasing at this rate.