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Question 3 (8 Marks) An artery of a flow network is al edge whose deletion causes the largest drop in the value of maxim flow . Let fbe an arbitrary maxim flow For ...

Question

Question 3 (8 Marks) An artery of a flow network is al edge whose deletion causes the largest drop in the value of maxim flow . Let fbe an arbitrary maxim flow For the following statements. prove them being correct 0r disprove them by finding counterexample. (a) An artery is ahl edge with the maximum capacity (b) Deletion of an artery decrease the value of the maximum flow by the capacity ol €

Question 3 (8 Marks) An artery of a flow network is al edge whose deletion causes the largest drop in the value of maxim flow . Let fbe an arbitrary maxim flow For the following statements. prove them being correct 0r disprove them by finding counterexample. (a) An artery is ahl edge with the maximum capacity (b) Deletion of an artery decrease the value of the maximum flow by the capacity ol €



Answers

Blood flows more slowly in the arterioles than in the artery that supplies them because the arterioles a. have thoroughfare channels to venules that are often closed off, slowing the flow of blood. b. have sphincters that restrict flow to capillary beds. c. are narrower than the artery. d. collectively have a larger cross-sectional area than does the artery.

This exercise. We have an artery that decreases in its diameter from D to the prime. The prime is 75% of D. So the private 70 0.75 D and in question A, we have the develop metric flow rate of the blood is the same in both situation. So Delta neither the t easy go to doubt of the prime. No, the tea. And we want to calculate how the pressure the difference in pressure in in the artery changes. So once you calculate Delta P prime, that's the difference in pressure after the decrease in the diameter of the artery as a function off the people. That's the difference of the That's the difference between off of the pressure before the decrees. So what I'm going to do is to use coarse as law which says that the volumetric flow rate is even by high over age times. Don't the p over l Aita l is the length of the artery. Aita is the disgust of the blood times the radius of the artery to the fourth and I'm gonna isolate about a P here to get eight over pie Aito l over our to the fourth daughter V though the tea And when I write it just little differently as eight. Tito el over high ah dot of either the tea TV. No, the tea all of these times went over are to the fourth notice that in both situations we have both before and after the decrease. This I wanted to hear 88 l over pied out of, you know, the tea is constant. Okay, so I'm gonna just call it, see, and we're gonna have that. The pee equals constant over our to the fourth. Okay. S o, this is the very of that API. That's before the decrease in the diameter. And let's calculate that a p prime, that's a p prime. It's gonna be see over 0.75 are to the fourth. Noticed that since the diameter was decreased by said that by 25%. Then so was the radius. This is gonna be 3.16 see over our to the fourth. But see over our to the fourth is Delta P is the original difference in pressure. So this is gonna be three points. 16 belt api. Okay, so we have the results that that'll be prime is 3.16 about a P. Or in other words, if we want the the percentage of increase, the difference in pressure was increased by 216%. Okay, now we move on to question V. And here, uh, were asked what happens if the difference in pressure was kept constant so that the P prime equals that api? What happens to the volumetric flow rate? So now what we want to do is to calculate Delta v Prime, not a t as a function of the outer vdot. Okay, So, uh, again, I'm gonna use pose as a creation. I'm just gonna write it, uh, again here, which is higher age that a p over L Aita are to the fourth. So now what's constant? Is this guy here that I wouldn't call D? Okay, this is a constant. So delta V that a t is de times are to the fourth. Okay, this constant, because the peace constant, Ellis Constant and Aitor is constant. So we have that doubt, Avi, though the tea is de are to the fourth. And this is the initial volumetric flow rate. The final volumetric flow rate is gonna be the out of you primed at the tea, which is D times 0.75 are to the fourth, which is 0.32 d r to the fourth. And since d r to the fourth is that I ve delta t We have 0.32 and out of E. Delta t case of the results, uh, is here that if you prime equals dealt it, that would be a prime over that. A t is 0.32 Delta veto the tea. Yeah, if we want the decrease in in percentage 81 minus 0.32 0 When? 68. So the decrees is 68% and this concludes there exercise.

Okay, so we're given arm function to be of our on for part A. We want to have oh, it V when X is equal to or or is equal to 0.1. So we'll display this into our equation here and now. Let's put this into our calculator. So they're 18 500 then points to five months 50.1 square. So this is equal to 44 40 and I'll be of 0.4. That's one it 500 times 0.25 minus 0.4 to depart to. So it's well this into our calculator. So this is equal to 1665 OK, so let's notice here, this is our radius. So when our radius is smaller, you see that V of R is larger, okay, and then when are is large, we see that the of our it's small and now for port. See, we want to create a table for our values. We want our you believe your own to appoint one 0.2, 0.3 and 0.4 as well as 0.5 So just going that into my calculator and we'll find our course wanting to be a bar, so we have zero. So that's 46 to 5 and inhale 50.1, that's 4440 and then we have point to That's going to be 38 It's five, and in 50.3 that's 26 or exit its nine 60 and input for that's 1665 and finally 16650.5, that's going to be a zero.

So in this problem we have a pressure difference in the artery of giving which is 150 Newton. For neither sprayed from example. We've been then. And the arteries religious of 2.4 millimeter a length of 8.5 centimeter. And uh conversion of viscosity of blood is mhm 2.72. and stand this negative three Newton 2nd per introspect. Now Question eight. Us for the volume of the blood that flows per second. So this is the blood flow rate. Okay or the volume flooring of the blood. We can mr Mendez from classes equation. Mhm. Mhm mm. Okay. Rich. The volume of the blood so great. It's equal so this. Mhm. Yeah. Yeah. Yeah smoking Certainly given values. You have 2050 paschal's and you tell parameter spread to the pressure difference in their ages. That's been very Millimeter two m. We have 0.00 24 L 1-4 over. It's kind of the eight notified by 2.72 times. Then it's negative three Newton. Second parameter spread multiplied to length. Let's convert centimeters to meters. We have 6.0 85 meters. Mm solving We can find that this is equals two 2.54. and standards the -5 m Cube. Where second or 25.4 cm two for a second. Since you just have to notify 100 basically don't work. Computer Cube. Now question be us if the rages is 18% less than that given. So that's the notice new. Did you start to in the given as our one day 18% less? So 119, So we have 82% of are women. It's the size of by what factor of the blood flow away is reduced. So what are probably asking for inspiration? The given blood flow rate Over the new 1? So this is what then while everything is have the scene. So searching the formula. Okay. All right. Mhm. Yeah, mm. Since they are the same this up and I wish to reduce this to this And we know that our two is 82% of our once we have zero points 82 Race the four and our oneness before Solving this is equal to one over 3.82. Based on four, it's equals two. Three point as such. That volume Florrie is reduced by a factor of two point.

So for a party, the volume floor it He's in quote too. The area times me. So given that the volume flow rate is 3.6 times 10 to the negative six, that will be going to the area off the artery which will be pi r squared. And the R is my 0.3 times 10 to the negative, three meters squared and then you're gonna find the so solving this for me is 30.42 meters per second. Now, for part B, we're gonna say they want to be one is equal to A To me too. Now a truth is actually equal to 1/3 because it says that the artery is constricted. So we have one times 0.42 is equal to in 1/3 times the to weaken by across the stevens and that's when I gave me V two is equal to three times you know, points, you know, for two. And that equals your 20.1 to 6 meters per second


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