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Problem 19 Hard Difficulty

The rate of change of atmospheric pressure $ P $ with respect to altitude $ h $ is proportional to $ P, $ provided that the temperature is constant. At $ 15^o C $ the pressure is $ 101.3 kPa $ at sea level and $ 87.14 kPa $ at $ h = 1000 m. $
(a) What is the pressure at an altitude of $ 3000 m? $
(b) What is the pressure at the top of Mount McKinley, at an altitude of $ 6187 m? $


a) $64.5 \mathrm{kPa}$
b) $39.9 \mathrm{kPa}$

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Video Transcript

in this problem. We're told that the rate of change of the atmospheric pressure with respect to altitude, which we would describe as D p. D. H is proportional to pee and is proportional. The P means it equals some constant K Times peak. And whenever we have a relationship like this, what we've seen in this lesson is that it translates to the exponential growth and decay equation, which would be P equals p, not times e to the k times h so we can use this equation to solve the problem. In addition to the initial conditions that were given. We're told that at a height of zero, the pressure is 101.3 Killer Pascal's and we're told that at a height of 1000 meters, the pressure is 87.14 Killer Pascal's. So let's use that information and find the value of K. All right, so we're going to substitute 87.14 in for the final pressure and 101.3 and for the initial pressure and will substitute 1000 in for the height and we'll solve this for K, so we'll divide both sides by one a 1.3, and as a fraction that ratio is going to simplify to be 4357 over 5065. So that equals E to the 1000 K. Then we take the natural log on both sides, and then we divide both sides by 1000 and will have the value of K. All right, now, this is not a very user friendly, user friendly number. It's around negative 1.5 times 10 to the negative fourth. So what I decided to do is store this in my calculator so that I don't use any so that I don't lose any accuracy when I go to use it again. All right, so now we want to use that value in our model, and our model is P equals one. A 1.3. The initial pressure at zero degrees of altitude, times E to the K. I'm just gonna call it K. For now, we know the number, but I don't want to write that number every time. So K times h. So let's use that model to do parts A and B. So for party were finding the pressure at an altitude of 3000 meters. So we substitute 3000. And for H, we're going to use the K value that was stored. Put that in the calculator and we get approximately 64.48 killing Pascal's. Now. The book answer is very close. They have 64.59 could be that a rounded value was used to get the book answer rather than the exact value that I used. So I suspect that mine might be slightly more accurate because I didn't round anything. Now we're gonna go on to Part B and we're doing the same thing. But we're finding the altitude. We're finding the pressure at an altitude of 6187 meters, and that would be the top of Mount McKinley. So we substitute 6187 in for the height. Use your stored value of K, put that in the calculator and we get approximately 39.91 Killer Pascal's of pressure and the book has 40.5 again, probably just a rounding thing