An ideal gas is compressed and expanded in such a way that its temperature remains…

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

An ideal gas is compressed and expanded in such a way that its temperature remains constant, so the product of its pressure

P

(in kPa) and volume

V

(in m

^{3}

) is constant. Answer the following questions about the moment when the pressure is observed to be decreasing. (a) Explain qualitatively how the volume must be changing at that instant and why, making explicit reference to the relationship between

P

and

V

. (b) At that instant the gas has pressure

P=200\text{ kPa}

and volume

V=0.010\text{ m}^{3}

, and the pressure is changing at a rate of

\dfrac{dP}{dt}=-5\text{ kPa/s}

. Without calculating an exact numerical value, determine the sign of

\dfrac{dV}{dt}

and compare its magnitude to that of

\dfrac{dP}{dt}

. Support your answer with mathematical reasoning that follows from the condition

PV=\text{constant}

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