Question

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 630 cubic centimeters and the pressure is 89kPa and is decreasing at a rate of 12 kPa/minute. At what rate in cubic centimeters per minute is the volume increasing at this instant? cm^3/min​ (Pa stands for Pascal -- it is equivalent to one Newton/(meter squared); kPa is a kiloPascal or 1000 Pascals.)

Answer 1

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 630 cubic centimeters and the pressure is 89kPa and is decreasing at a rate of 12 kPa/minute. At what rate in cubic centimeters per minute is the volume increasing at this instant? cm^3/min

Step-by-step explanation:

89 kPa * 630 cm^3 = C

C = 3.54885 x 10^9 cm^4 kPa^(-1.4)

Now that we know the value of C, we can differentiate the equation with respect to time. This gives us:

dP/dt = -12 kPa/min

dV/dt = (1.4)(C)(P^(-1.4))(dP/dt)

dV/dt = (1.4)(3.54885 x 10^9 cm^4 kPa^(-1.4))(89 kPa)^(-1.4)(-12 kPa/min)

Therefore, the rate of change of volume is 910.846 cm^3/min. This means that the volume is increasing at a rate of 910.846 cubic centimeters per minute