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Tadine BennsttSa edtcthh PCenfnteia Pretected 329 6010 Jlua *(orMnl Irv ~Wbl LdugMldailLi ( BntanvninWL hiom INamttrrunsferred 10 (miversity hospital from week okd ...

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Tadine BennsttSa edtcthh PCenfnteia Pretected 329 6010 Jlua *(orMnl Irv ~Wbl LdugMldailLi ( BntanvninWL hiom INamttrrunsferred 10 (miversity hospital from week okd nule ww WLS Patient Atls The child" $ spells begon I0 (kry hislory ol choking spells Stholl CUmHty hospitol with= brerth; In Ihe pror doys,he anll coughg and progressed to his (sptng for NTemive= spells: His physical mth of vomiting assaciation with hs choking akso Iad three "Tsodes = an] respiratory rate of examimtion Was s

Tadine Bennstt Sa edtcthh PC enfnteia Pretected 329 6010 Jlua *(or Mnl Irv ~ Wbl Ldug Mld ail Li ( Bntanvnin WL hiom I Namt trrunsferred 10 (miversity hospital from week okd nule ww WLS Patient Atls The child" $ spells begon I0 (kry hislory ol choking spells Stholl CUmHty hospitol with= brerth; In Ihe pror doys,he anll coughg and progressed to his (sptng for NTemive= spells: His physical mth of vomiting assaciation with hs choking akso Iad three "Tsodes = an] respiratory rate of examimtion Was sigmficant tor pulse of 16O heats per There was no evidence of Tracheal The child'$ chest radiograph Vs cleur . 72/min Culture from nasophr-geal swab His wlute cell count Ws 15, 500/(. abnoralities revealed sll Gram ~negutive bacillus ANSWER identifying this infection disease and the infectiaus agent , In Addition to THE FOLLOWING QUESTIONS: Were this child' clinical course and chesl radiograph consistent with his infection? Explain How might this disease be prevented? What therapy would YOu suggest for this patient? Clinically , the cough iay persist tor some time following therapy Give possible reasons why cough may persist In the face of therapy? 0Focus



Answers

Coughing When a foreign object lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction of the trachea, making a narrower channel for the expelled air to flow through. For a
given amount of air to escape in a fixed time, it must move faster through the narrower channel than the wider one. The greater the velocity of the airstream, the greater the force on the foreign object. X rays show that the radius of the circular tracheal tube contracts to about two-thirds of its normal radius during a cough. According to a mathematical model of coughing, the velocity $v$ of the airstream is related to the radius $r$ of the trachea by the equation
$$v(r)=k\left(r_{0}-r\right) r^{2} \quad \frac{1}{2} r_{0} \leqslant r \leqslant r_{0}$$
where $k$ is a constant and $r_{0}$ is the normal radius of the trachea. The restriction on $r$ is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than $\frac{1}{2} r_{0}$ is prevented (otherwise the person would suffocate).
(a) Determine the value of $r$ in the interval $\left[\frac{1}{2} r_{0}, r_{0}\right]$ at which $v$ has an absolute maximum. How does this compare with experimental evidence?
(b) What is the absolute maximum value of $v$ on the interval?
(c) Sketch the graph of $v$ on the interval $\left[0, r_{0}\right] .$

For the volume problem, we want to consider a foreign object lodged in the trachea. We get the mathematical model of coughing and give me the velocity of the air stream. So that's velocity because the function of the radius of the wind pipe that skank. Okay. Times are not mine are, times are squared. What do we have are not as the normal radius of the trachea. Um So what we want to determine is the value in which there is an absolute maximum. So we take the derivative, we prime of our and we would end up getting um we want to use the product rule or we can multiply this through because we have K. Are not R squared minus are cute. I would rather use this now because then we don't have to use the product rule and that makes it much easier to find where it's equal to zero.

So for this question, we're trying to maximize the velocity V off the wind coming out of our windpipe. Given the radius R and radio artists between must be between when half roo roo roo being the normal radius of our windpipe. So in order to do so, we need to do is take the derivative of the We're gonna be crime of our That's gonna be equal to SUNY Do channel to this. So we're gonna have if we drive, the armaments are moving again Negative k r squared because the arm eyes are just going to drive to native born right and then we're gonna have plus a two k r r minus are that's gonna be equal to zero. And then next time I just moved accounts great to other side somehow to k r Cindy grocery. I'm sorry. Two kr our monies are is going to go to K R squared one of ours we can't solve in the cave can solve. So get to our Marlys to our it's getting equal to our to our is gonna be good If the are or ours only go to 2/3 are oh, so this checks out with the experimental value given to us home earlier in the question. Okay, So for part B, we're trying to find the absolute natural value of the So when you do is playing in this 2/3 roo until RV equation. So we're gonna have the u 2/3 are. It's gonna be equal to K R O minus 2/3 are, and then 2/3 are square. That's gonna be equal to que times, um, are almost to tears. Roo is gonna be 1/3 are terms 4/9 R squared. Okay, most flying this out. We're going to get that. The of to 2/3 are gonna be equal to four k arl to the third over. Ah, 20 seven. Alright, India, That should be our answer for that Velocity V. Now, for part C, we want to sketch a graph of e on interval of zero to r O. So we're gonna start here is zero and then are looking had to urge our somewhere in between. Right then we know we're gonna have a maximum there for V given the tutors are and let's check out what V looks like when R is equal to zero. And when our Zoro so rz Quito zero, we're gonna get the, um Okay. Harlow minus zero and then zero squared. So this is gonna be zero v of tomorrow. It's gonna be ok. Are my eyes are oh, desk and just be zero. So, for both these points, mathematically speaking, we would get zero and graphic looks something like this with a maximum value to their zaro.

So in this case, it really depends on what the constant is K. We see that if it's lesser, we won't have is larger maximum. But if it's greater than we'll have a larger maximum value, we also see that it will depend on the value of our not, which is also variable, that would dramatically change the value of the maximum. Um, so in this case, if we just increase both of these, the absolute maximum would be, in a sense, infinite, which does not really compare with experimental evidence because we know that there is a limit. Two are not. We know that there is a limit to the radius of the trachea, which we are not provided with. Um however, if we want to find the absolute maximum of the on the interval in this case, uh, this is our interval. Um, we see that the absolute maximum would be two thirds, but again, that could change. Um, as we increase the certain values so example, if this went to to we now see that this would be still at two thirds, but the maximum would be higher. Um, and then if we increased this from 1 to 2. We see that goes from two thirds to, uh, 4/3. So it doubles eso we see that that's how that all works. And then if we sketched the graph so this is gonna be from zero to our knock. That would be this graph right here and again. It depends on the value of our not and K, but we do see that it follows appropriately.

Hey, guys, lets troll Problem 41. The first question is wife was Gene is an effective weapon on low spots. We know that the density off for spring is 98 19 to Grandpa Mall and the density off air is 29 g per mole. Therefore, we see that fortune is a lot heavier than air and the Russian off the DNC disease 3.41 we stay. This pressure is higher. More than one. Andi Wasin is a lot heavier than air and it will displace the air near the ground, such as in the trenches on the ground. Let's try to this town. It is a written here and in this place, Yeah, near the ground. And for this reason, it is an effective weapon. Okay, Next we need to calculate how many Grandma's off. Austin. He's in the feet sample. We know that volume off Cylindrical Tubize. It is my our square age. And now it is given that the outside diameter of the do is there 135 centimeter. The one thickness given is 0.559 centimeter, 0.559 centimeter. Therefore, the insect damage in or the two wins. 01 635 minus the wall thickness. Want to play the brain to which gives the inside diameter off that your best 0.5 to 32 centimeter different? The volume that you bees, The height even is 15 centimeter and the volume of the day break it is 3.22 Sentimentally square centimeter cube. The volume of the Cube. Now, if the plant had warned, then the amount off Gina would have been introducing the guest. Photography's the amount off or swim. Let's get a new page then they want a phosgene introduced into the gas chromatograph. Um, he's the mess we know message can to volume multiplied. Viridian City Before we can write, we know density is a little pressure multiplied by Muller mess divided by Opti. Now we can insight on the gallows here the volume off the two with 3.22 centimeter cube. Let's come back to Leader, my dividing by 1000 centimeters. The pressure given here is one in him. The Mueller mess so far Genius. 98 point 92 grand Permal. That's constantly 010821 liter a team morning, your son Kevin Anderson and the temperature even. Here is 23 Calvin Ex con wanted to 2032 centuries that's converted to kill them by adding to 73. And when we do this calculation, we get the Grandma law for swimming in the fifth Central on to promote the Grammys. 0.1313 Graham. This is down Grandma's off Austin in the fifth sample. Mhm in the first sample. The next problem is or to the safe concentration off Jean exceed. If all the question evaporated into the room, we need to do some calculation. Transferred this question. No Dumb Moon's off oxygen in the air the number of months and is built to massive aided by Mueller Mess. The mess we just calculated is 3.22 Sentimental Cube. Let's converted program. We want to play in 1.37 and the Mueller mess is 98.92 Grandpa on one. And when we do this calculation, we get the most off US win in the area. 00446 moments. Now the most off air we can can collect the months affair we can use the item gets the question and is going to be V by rt The pressure given use one. It was very Christian. The volume ease given us 2200 Fit cube that's converted to leader by multiplying 28.317 Leader Yes. Constant is 0.0 8 to 1 leader. Administrative pressure. Mullen, your son Calvin in your son, the temperature is 23 d uses, she s and we get to 563 mol affair. Now we can calculate parts more million off oxygen in the air, parts per million off phosgene in There is a question number of months off Austin in the air and number off most off Yes, which gives 17.4 months ago by 10 in over six or we can say 17.4 p p. M. Now the parts per million off question in the air is more than the safe level, off 0.1 ppm. Therefore, question would have exited the 67 However, even in the question where below the self level, there would still be an answer. Live in the end, the floor because fortune is denser than air and will displace it. The next Russian is that things we did that made the experiment unnecessary has under what of the things that he would have done. So the things we did which made the experiment has a disease. The first thing we did not follow the proper safety protocol district in the material safety data sheet and fortune it needs for adequate personal protective equipment such as Gas Mosque and the experiment would have been done in a film head. As a precaution, he should have followed thumb safety protocols and product have used. He should have used from the productivity movement so he could have done these things to make the experimentalists hazard us.


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