How Fast Does Volume Increase When Air Expands Adiabatically?

[shaunanana]

Homework Statement

When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV^1.14=C, where C is a constant. Suppose that at a certain instant the volume is 600 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?

Homework Equations

dP/dt=-10
we want dV/dt when V=600 and P=80

The Attempt at a Solution

V=(1.4 root)(C/P)
80(600)^1.4=C
C=620157
dV/dt=1/1.4(C/P)^-0.4((cp'-pc')/p^2)dP/dt
=1/1.4(C/P)^-0.4((c10-0)/p^2)(-10)
then i plugged in P=80 and C=620157 to get an answer of 192.5 cm^3/min which was wrong.

Can anyone show we where I went wrong and how to get the proper solution?
=

 

[Mark44]

Homework Statement

When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV^1.14=C, where C is a constant. Suppose that at a certain instant the volume is 600 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?

Homework Equations

dP/dt=-10
we want dV/dt when V=600 and P=80

The Attempt at a Solution

Error in next line. The constant is 1.14, not 1.4. Also, there are square roots, cube roots, fourth roots, and so on, but not 1.4 or 1.14 roots.

V=(1.4 root)(C/P)
80(600)^1.4=C
C=620157
dV/dt=1/1.4(C/P)^-0.4((cp'-pc')/p^2)dP/dt
=1/1.4(C/P)^-0.4((c10-0)/p^2)(-10)
then i plugged in P=80 and C=620157 to get an answer of 192.5 cm^3/min which was wrong.

Can anyone show we where I went wrong and how to get the proper solution?
=

PV^1.14 = C, so V^1.14 = C/P, so V = (C/P)^1/1.14