Answer:
The gas is decreasing at a rate of 50 cubic centimeters per minute.
Explanation:
The Boyle's Law is represented by the following expression:
[tex]P\cdot V = k[/tex] (1)
Where:
[tex]P[/tex] - Pressure, in kilopascals.
[tex]V[/tex] - Volume, in cubic centimeters.
[tex]k[/tex] - Proportionality constant, in kilopascal-cubic centimeters.
By definitions of rate of change and implicit differentiation, we derive the following differential equation:
[tex]\dot P \cdot V + P\cdot \dot V = 0[/tex] (2)
Where:
[tex]\dot P[/tex] - Rate of change of the pressure, in kilopascals per minute.
[tex]\dot V[/tex] - Rate of change of the volume, in cubic centimeters per minute.
Then, we clear the rate of change of the volume within (2):
[tex]P\cdot \dot V = -\dot P\cdot V[/tex]
[tex]\dot V = -\left(\frac{\dot P}{P} \right) \cdot V[/tex]
If we know that [tex]P = 180\,kPa[/tex], [tex]\dot P = 30\,\frac{kPa}{min}[/tex] and [tex]V = 300\,cm^{3}[/tex], then the rate of change of the volume is:
[tex]\dot V = -\left(\frac{\dot P}{P} \right) \cdot V[/tex]
[tex]\dot V = -50\,\frac{cm^{3}}{min}[/tex]
The gas is decreasing at a rate of 50 cubic centimeters per minute.
