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Problem 37 Medium Difficulty

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 constant, 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?

Answer

The volume is decreasing at the rate of 80 $\mathrm{cm}^{3} / \mathrm{min}$

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Video Transcript

were asked to answer a question about Boyle's law using related. So Boyle's law states that when a sample of gases compressed at constant temperature pressure p volume v satisfy the equation. P Times V equals C. Percy is constant said that was like and the pressure is 150 kg. Paschal's mhm and the pressure is increasing at a rate of 20 killed. Paschal's per minute hardcore history. Dp DT is 20 polity Asians and were asked at what rate the volume is decreasing at this instant. Find this. We're going to differentiate the Boyle's law equation with respect to t buy the product rule we have dp DT Cam's V plus p times DV DT On the right side, we simply have zero. C is a constant solving for d v D t. Is it? Yeah, this is equal to negative v overpay times stupidity. So what? And plugging in our values We get negative well defined. He he want to plug in 1 50 to our equation. So we have 1 50 v equals sees that V is equal to see over 150. So this is negative. See over 150 over 150 times the PDT, which was 20. So this is negative. 20. See over Say yeah. Wants using said you know leads combat Yes, yes, over 1 50 squared space. Perhaps you're missing an important piece of information in the problem. In the book, we're told that the volume is 600 cubic centimeters. So actually our volume is not see over 1 to 15 just saying but 600 sticks and pebbles everything. So our rate is now equal to negative 600 over 150. I was 20 which is negative E and the units of DVD T. These are cubic centimeters have per minute. We have a negative sign, which means that the volume is decreasing and we had that the volume is decreasing at a rate of 80 cubic centimeters per minute. Just like