a-find-the-general-traffic-pattern-in-the-freeway-network-shown-in-the-figure-flow-rates-are-in-cars

Question

a. Find the general traffic pattern in the freeway network shown in the figure. (Flow rates are in cars/minute.)
b. Describe the general traffic pattern when the road whose flow is $x_{4}$ is closed.
c. When $x_{4}=0,$ what is the minimum value of $x_{1} ?$
figure can't copy

Runpeng Li · Accepted Answer

Okay, So in this problem, we need to describe the general traffic pattern or to give them figure. Now, we just need to make a table out of these four intersections and with the flow kinda blowing and the flow out So here we have a B C D for intersections and flowing and blow out. Okay, So for the intersection, eh? We have X one closing and 40 last x three less x four blowout. And this should be equal or intersection B. We have 200. No wind and X one A sex, too. Blowout intersections. See, we have x two. And next three go in and 100. Uh, X five go out or into section D. We have X four in next five. Blow in and 60 comes out. Okay, So based on this table, we can make a matrix. That is one. Uh, we this denote the first column as ah X one. Next to x three. It&#x27;s four. Ex fight. Okay, So the first question we have, uh, one of x one and put the x three from the right hand side. You left inside. So that is negative. One for detective for X three and it&#x27;s the same four x four we have next one and forties Constant out there. So this should be 40 and one of x one and neck three connected. One, Rex three and export X three x four. So arrested will be zero. Okay. For the second or the second row, we have 11 and all zeros 60. Awake should be 200. Sorry. 200. And for the third row at 011 because we have one of ex to one of x three. And next one off expire next while backs five. And the council terms is 100. Excuse me. Okay, So for for the last roll at all zeros, we only have one of exploring the wild. Well, enough, expert. So that&#x27;s 11 and constant term is 60. All right, so we can perform the cashing illumination to this matrix. Um, to do that, we first used a second row. Subs tracked first roll. So that is 100 Sorry. 10 connective 110 and working. So one minus one zero one minus zero. One on this 3110 and 1 60 110 1 60 okay. And keep the third row. And fourth rule 100 worker will be three zeros. Scuse me. Three zeros and too once and constant term is 60. Okay, the next we used a third row. Subtract a second role. So we first set down the first roll next to one negative one zero or tea. 0111 zero 1 60 Okay, um so third row, subtract second role. That is 00 Next one, a negative one and 60 connected 1 to 1. And like their 60. All right on we keep the nice roll. 11 16. All right, so right now, right now, we can First officer of the third row on the fourth, who will be the same. So fourth row will be cancelled. So that is okay, we have 10 notice. 1 91 zero 40 and 10111 Do? Yeah. 1 60 and 00 You know, 11 and 16. The last room is all heroes. Okay. Finally, we used a second rule. Subtract the third row. So this gives us first one. Do active 110 and 40 on dhe. Second role will be your 211 So one minus one, which is zoo on zero minus one Negative. One 1 16 minus 60 will be 100 and the third rule will keep the same. Oh, that is very little. 011 and 60. Yeah, 60. All right, So you&#x27;re Here&#x27;s our reduce station of form. Now we plug in our variables. That is x one minus x three minus x four is 20 and x two plus x three minus x five two plus X three minus X fly. It&#x27;s 100 all right? And the last one have X floor, plus x five is 16 so x five will be our will be the free bearable here. Oh, right down x one turns off 40 plus x three US explore and next to will be negative x three. Plus it&#x27;s five and x three is a free bearable. Well, that&#x27;s four is 60 financed X five. So here we have 23 variables X three, x five and this is our traffic pattern. Okay, Um so for the second problem, we need to describe the General General Patrick of Pattern. When the road who&#x27;s flow is exports closed. So that means we need to said x four as zero, because that&#x27;s four is closed. So that means for according to our the fourth equation here, so enemies, our X five is 50. All right, so Corey&#x27;s found early. We have Thanks. One is 40 plus x three. Thanks for. So this will be 40 those x three and next 480 And secondly, X two will be connective X three, class 50 because x five here is 50 and ex threes. Steel of free variable. This won&#x27;t be changed, but X for his ex warrior zero and the last. The NASA question turns out to be x five People&#x27;s 50. I&#x27;m sorry. 60 60 here. Yeah, X files should be 60 because x four here, we&#x27;re assuming x 40 So x five should be 60. So I have to revised the value of X fire here. So that is 60. So it is 60 and next to is equal to negative extra plus x five so that this should be 60. All right, so here&#x27;s the traffic pattern for our 2nd 2nd question. So for the last question, we have you export zero, then what is the minimum? Better off X one. So, for the quick first question here we have, we&#x27;re we have X one equals 40 plus x three. So this is under the condition affects four equals zero. So we the thing we need to consider here is to minimize x three so that our RX one here will be the minimum. So in order to meet him, minimize x three. The best we can do is to set x 30 So x one will be the month, which is the meeting will be 40. And here&#x27;s our solution, while for the second question and we&#x27;re done.

Ankit Gupta · Answer

um, I River personage do a decent The fracture on multiple Gerson or Paula meals already result in ethical normal. That&#x27;s the reason why owned by no good man for Paul. Normal is if you do a decent direction about the Lakers and then already are getting all normal. But if you divide, then also really grateful, normal or no? So what? I can do a human problem in your exit door. Let&#x27;s say my next is great. Let&#x27;s it text. Bless on a little number 11. Considering your exit. Thanks for calling. Why? No, you fight your facts and your pics. Then what? And yet your pecs plays. You are exported fire then simply near where I support five year also, so we&#x27;ll be right in order. Then I&#x27;ll get a support. Priceless. My nexus cried. Plus six x plus three. Getting parliament, then personable before sub flexes alcohol and sample multiplication else. But this will be wired in what you know the office unitedly if they find your picks in the Bible. My nexus bread. It&#x27;s explicit. E my dear Agnes, one by two affects. That&#x27;s fine. There is a regular stumps. What? I love my nexus. My divided about you. It&#x27;s a bar fight, less sexy writer by It&#x27;s a bar fight anybody but you. It&#x27;s born by. So if this is divided Sindicato water no power there and five power years so we will be left on my divided by I love excuse. Let us know what everybody it&#x27;s about board unless they had a really want her life and quality calculated, isn&#x27;t it? Is all Assad and negative now? This simplified. Then it&#x27;s what three dick in numerical advantage. With their support minus city less, it will basically part minus four. Listen to the finish with one minus fight we can take here. Follow Assad in negative. So far for Longyan also mean so not been negative numbers so they can take when the divide. Then we won&#x27;t ghetto available longer conventional get and sometimes known if degrees off. Notice this Tinto General for longer. But this is not getting so like that is the reason That&#x27;s what we are not getting always for London division. Thank you

Runpeng Li · Answer

Okay, So in this problem, we need to find out the find the general solution of the network flow. And here we have six intersections that it&#x27;s a B C D E f, and we need to counter flow in and blow out. Okay, so for Intersection A, we have X one closing on next to does 100 blowout. And for Intersection B, we have X two plus 50 No win and next three goes out. Now we&#x27;re into section C. Where have X three blows in and X four. That&#x27;s 100 and 20 no out into second D Explore and 1 50 slow win and x five goes out or into section E. We have X five close in and X six bus. Hey, t blowout! They&#x27;re in two sections. Have we have X six last 100? No win and X one flow out. Okay, so all these expression should be equal. And so based on our table here, we can write down corresponding matrix. So noticed that this should be a six by should be a six by seven matrix. So excuse me. No, that&#x27;s the note column, SPI X one X two X tree Teoh X six for bye six. And for the first for the first roll, we have one neck of one and other article you visions will be zeros. And the council term, it&#x27;s 100 for the second row. We have one for a X to neck of one breaks three and other other terms are all zeros. So on the cast in terms next 50 and their role similar T constant term is 1 20 and fourth row would be one negative one and constant term is negative 1 50 and they throw one next one and 80 The last rule we have connected one or x one and one for X x six and accounts in terms of connective 100. So all in terms, art zeros. So here&#x27;s our metrics. Now we need to apply gushing gushing elimination. Uh huh. Scuse me, um, to get to get the reduce station for me. So in this case, I would just just write it down. So it is 11 111 We have five ones and zero, and these terms will be next one Next one. Next one, Next one. So we have five negative ones, and it&#x27;s all Jews. And the last last column we have under zero 50 connective 17 Amy and New. Okay, so based on our reduced Asian of form, you know that X one minus x six is It is 100 and next to minus X six is zero x three minus x six iss 50 Next four minus next six is note 20 and x five. My list six is 80 and next six is pre variable. All right, so we stopped the first question. Now the second question is to find the smallest possible battle for X six. So in this case, um, we need to find XXXIX as small as possible, but noticed that ax to is Symantec&#x27;s six. If we take XX as that&#x27;s my password, that C zero, the next to will be zero and the rest the resident. The rest of variables will be x one is 100 x two zero x three 15 x four Connected 20 and x five 80 now Excuse me. So we don&#x27;t allow negative number for for the flow because now we have we take x x six as zero. Then export will be next 20. This this is impossible. So in order to make a and this non active a negative X War. So we need X six to be at least, um, Daddy&#x27;s we need to make exit X four minus x six, because next 20 your ex for is negative 20 last x six. So, Dad means we need to make X six minus 20 bigger or equal to zero because export cannot be cannot be negative. So that means X six minus 20 should be bigger. Only 06 you going to 20. So put in for the from the mean minimum of six, we can take xx to be 20. This this is our final solution for the minimum xx we can take and we&#x27;re not.

Evelyn Chen · Answer

for this problem were asked to find the largest possible value of X three. As you may know, the total flow into a network flow should be equal to the total flow out of it. You may notice that all of the branches connected to note CR flowing out of it. This is a mistake by the publisher in the branch label. 80 should be flying in to see. To solve this problem, we must first find instead of equations for each note, plus an additional equation for the total for no day. Excellent and x three are flying in while 20 is flowing out. This makes the equation. X one plus X three is equal to 20. For now would be you&#x27;ll find that X two is flowing in or x three and X for flying out for node. See eighties hauling in or excellent and next you&#x27;re flying now for the toil X four and 20 are flowing out. What 80 is flying in once you have. This is these equations. It&#x27;s best to put all the variables on one side of the equation and the numbers on the other side to make it easier to make the matrix for the first equation. The variables are on the left side. Well, the numbers on the right side. If we follow this pattern for B, the equation becomes x two minus x three minus X four is equal to zero for C. The equation becomes X one, plus X two is equal to 80 for the total. The equation becomes X for is equal to 60 which is a T minus 20. Using these equations, we can create Our Matrix in the Matrix. The first four columns represent excellent x two, x three and X four, respectively. In the last column is the total. For example, for a no day, which would be Row one. The first column is X one, so that would be one for X to. There are no excuse in the equation, so it becomes zero for X three. That&#x27;s a one for X for that zero, and the total is 20 for be excellent. Zero x two is one, and because extreme x four are subtracted from X to, they&#x27;re both negative one and the answer. Zero. So that&#x27;s the last call for C. There&#x27;s one x one who are next to zero x three Xerox War and the tall is 80 for the last row or the total flow. There&#x27;s no X ones, no X twos, knocks threes, one x four and it always 60. Next, we must convert this matrix into row reduced. I shall want for to do this. I&#x27;m going to first attract Row one from Row three and put it in Row three. This means that the first two rows in the last row will remain the same, but the third row about every number in the third row, will be subtracted by the numbers in the first row. So what&#x27;s first right down the 1st, 2nd, 4th rose, As you may recall, the first row was 10 10 20. The second row was zero one negative one negative one zero, and we&#x27;ll come back to the third row in the last year was zero 00 1 60 The original third row was 1100 80. So we just need to subtract each of these numbers by 1010 20. So one minus one is zero. One minus zero is one zero minus one is negative. One zero minus zero zero a B, minus 20 is 60. Next, I&#x27;m going to subtract Row three and wrote to by Road to and put it in Row three again. I&#x27;m going to begin by writing. Read writing Rose 12 and four since Source attracting Row three from Row two. This time zero minus zero zero one minus 10 Negative one minus negative. One zero zero minus negative. One is one and 16.0 is 60. Here you&#x27;ll notice that the second to last row in the last row are exactly the same if you recall in row reduced echelon form. If there is a road that is completely zeros, it must be at the bottom. Therefore, we must subtract Row three from row for or row for from Rove three. Since they&#x27;re exactly the same and put it in row for this makes our matrix become 1010 20 01 negative, one negative. One zero 0001 60 and finally, the last year become 000 00 Next, I&#x27;m going to combine Row two in Row three and put it in Row two. For this I&#x27;m going to begin by writings. Rose 13 and four as they dont change for row to since riding road to and row three zero plus zero zero zero plus one is one zero plus negative one is negative. One one plus negative one and zero and 60 po zero is 60. Using this final matrix, we can find a solution set. This is some of equations is x one plus x three is equal to 20. We get this from the first row, X two minus X three is equal to 60 and we get this from the second row. And finally x four is ego the 60 when we get this from the third row? Because the final rose all zeros. There is no equation for that row. We can write these equations in parametric for X one is equal to 20 minus x three, and you can see that from the first equation for X two. It is 60 plus x three and you can see that in the second equation will come back the x three and X for 60 for X three. You&#x27;ll see that in our final matrix that there is no pivot point x three. This means that X three is a free variable. If you notice an equation, one X one is equal to 20 minus x three. But you know that X three cannot be greater than 20 or that would make X one negative. Therefore, the largest possible value of X three is 20.

Ankit Gupta · Answer

and it and intersex and costs that I&#x27;ve randomly at an average rate of particles. Eugene the Johnson it is given FX magic were to one minus beat about minor geo point fight. Thanks. I would estimate the likely would and prevented that at least one car into intersex and within we did excellent plus part find it acquired one minor speed about my Nigeria 10.5 into two, which is a court one mine. And you know what? This is one. This is due to a 0.6 30. So it means the 63% Johnson that at least one car, at least one car will enter intersex and intersex. Um, doing Hey, two minutes. Idiot. Would be graph have general 60. What happened to the like? Live with at least one card interned during the 60 minute video during the autograph. So this isn&#x27;t good off for happens, Roberto. One minus eight about minus zero point. Right x. So the sea either time in egregious. This is I am so as the time Integrys is the probability the probability in egregious So at the time in in pledges, this is probably the house when creatures and the wearability also increases and indigenes toe approach one poverty. Can you make someone great? That is, that is almost started in that at least one card will one car will and then the intersection doing it 16 years. Idiot doing 60 minutes idiot. So let it answer.

Cameron Bunney · Answer

whether in this video we&#x27;re going to look an application of systems of equations. Here is a diagram of the in section off Ah, uh, for one way streets. We&#x27;re looking at keeping the flow of traffic moving. So we need to take each intersection and at each intersection we need that. The number entering is equal to the number of leaving on. This is the principle will base further calculations. I&#x27;m with that in mind. If we look at the in section one entering, we have 182 100 giving us 380 leaving we have W and said So it&#x27;s sequitur w plus zed in section two entering, we have W plus X leaving. We have 400 plus 200. Which secret to 600 looking at in section three? Um, entering we have why, plus 200 and leaving. We have X plus 30 on that in section four. Entering. We have zed plus 70 um, and leaving. We have Y plus 20 tiding this up and we have W plus zed is equal to 3 80 w plus X is equal to 600. Um, X minus Y is equal to 1 70 on why minus zed is equal to 50. So here is our system of equations. Now we&#x27;re going to use size, part a. Now we&#x27;re going to use gas Jordan elimination to solve these. So don&#x27;t write this in gas form using variable order W X. Why, Zet? So the first equation is one 10013 80. Then 1100 600 one, minus one. Well, 01 minus one. Sarah, 1 70 There is 01 minus 1 50 We&#x27;re going to start by taking row one on replacing it by rail. One plus wrote for, um, By doing this, we can, uh, see what we find. Um, very one plus. So one person areas one Serapis Seri 01 zero plus one is 11 minus one is 03 80 plus 50 is equal to 430. All other rose remain the same. So just fill them in Now. Now, if we look at very one again, if we replace it by Row one plus row three. Yes. Um, something special happens. Um, one plus eros 10 plus one is 11 minus 100 plus 004 30 plus 1 76 100 It matches thesis on drone Oh, which, if we replace Row one goes to row one minus for two, we just get a row of all zeros. 600 there one minus 101 70 001 minus one 50. Now let&#x27;s look what information this tells us. The first row tells us nothing. The second row tells us W plus X is equal to 600. Or that W is a 600 minus X third row houses X minus Y is equal to 1 70 or the X is equal to 170 plus. Why, in the final wrote tells us why minuses ed is equal to 50 or that why is equal Teoh Zed plus 50 W is in terms of X x, in terms of wine wise in terms of set, and that is free. So if we just say that his T some real number, this sent houses that why is equal to T plus 50 with an X is equal to 170 plus Why so plus T plus 50 which is equal to t plus 2 20 But W 0 600 minus X, which is 600 minus 2 20 minus to which is 380 minus t. If we put that all together, we find W X. Why is that are equal to 3 80 minus t on, then to 20 plus t. Then why is 50 plus T ons that is just tea for tea? Some really number on This is our complete set of solutions. It forms a family of solutions. Eso there isn&#x27;t just one for every value of tea will find a new solution that will keep the flow of traffic moving. Now if we assume that due to construction reasons zed is fixed at 50 cars per hour, we want toe calculate what the flow of traffic will be at and between the other intersections. Since that is equal to T, this just means substituting in for T equals 50. So W is a 3 80 minus T, which is 3 80 minus 50 which is equal to 3 30 x is 2 20 plus t so to 20 plus 50 which is to 70 four. Then why is 50 plus T, which is 50 plus 50 which is equal to 100 and then Zed is just equal to 50. So these are the flow rates between, um, all of the intersections. Thank you.

Problem

a. Find the general flow pattern in the network s…

Problem 12 Medium Difficulty

a. Find the general traffic pattern in the freeway network shown in the figure. (Flow rates are in cars/minute.)
b. Describe the general traffic pattern when the road whose flow is $x_{4}$ is closed.
c. When $x_{4}=0,$ what is the minimum value of $x_{1} ?$
figure can't copy

Answer

a. $\left\{\begin{aligned} x_{1}=& 100+x_{3}-x_{5} \\ x_{2}=& 100-x_{3}+x_{5} \\ & x_{3} \text { is free } \\ x_{4}=& 60-x_{5} \\ & x_{5} \text { is free } \end{aligned}\right.$
b. See solution
c. See solution

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Grace H.

Catherine R.

Maria G.

Kristen K.

Absolute Value - Example 1

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a. $\left\{\begin{aligned} x_{1}=& 100+x_{3}-x_{5} \\ x_{2}=& 100-x_{3}+x_{5} \\ & x_{3} \text { is free } \\ x_{4}=& 60-x_{5} \\ & x_{5} \text { is free } \end{aligned}\right.$b. See solutionc. See solution

Linear Algebra and Its Applications

Grace H.

Catherine R.

Maria G.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Grace H.

Catherine R.

Maria G.

Kristen K.

a. $\left\{\begin{aligned} x_{1}=& 100+x_{3}-x_{5} \\ x_{2}=& 100-x_{3}+x_{5} \\ & x_{3} \text { is free } \\ x_{4}=& 60-x_{5} \\ & x_{5} \text { is free } \end{aligned}\right.$
b. See solution
c. See solution

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications