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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

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 600 cm3, the pressure is 150 kPa, and the pressure is increasing at a rate of 20 kPa/min. At what rate is the volume decreasing at this instant?

To determine

To find: The rate of change of the volume of the gas at constant temperature.

Explanation

Given:

The pressure P and volume V of the gas are related by the equation PV=C, where C be any constant.

The rate at which the pressure of the gas increases 20kPa/min.

Formula used:

(1) Chain rule: dydx=dydududx

(2) ddx(fg)=fdgdx+gdfdx .

Calculation:

Since the pressure P of the gas and volume V of the gas related by the equation PV=C , where C be any constant.

Since the pressure P of the gas increases with the time t.

Therefore the volume V of the gas deceases with the time t .

Since dPdt=20kPa/min

Find dVdt  when P=150kPa and V=600cm3.

Differentiate PV=C with respect to the time t.

ddt[PV]=ddt[C]VdPdt+PdVdt=0

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