Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation PV = C, where C is a constant. Suppose that at a certain instant the volume is 300 cm3, the pressure is 180 kPa, and the pressure is increasing at a rate of 30 kPa/min. At what rate is the volume decreasing at this instant?

The gas is decreasing at a rate of 50 cubic centimeters per minute.

Explanation:

The Boyle's Law is represented by the following expression:

(1)

Where:

- Pressure, in kilopascals.

- Volume, in cubic centimeters.

- Proportionality constant, in kilopascal-cubic centimeters.

By definitions of rate of change and implicit differentiation, we derive the following differential equation:

(2)

Where:

- Rate of change of the pressure, in kilopascals per minute.

- Rate of change of the volume, in cubic centimeters per minute.

Then, we clear the rate of change of the volume within (2):

If we know that , and , then the rate of change of the volume is:

The gas is decreasing at a rate of 50 cubic centimeters per minute.

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