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70.2 Assume there is nearly uniform light traffic on & semi-infinite highway. Sup- pose initially the traffic was exactly uniform p(x, 0) =Po, but the traffic d...

Question

70.2 Assume there is nearly uniform light traffic on & semi-infinite highway. Sup- pose initially the traffic was exactly uniform p(x, 0) =Po, but the traffic density was prescribed at the entrance, being slightly different from Po, P(o, Po + eg), (> 0, where 0 <eg() < Po: Determine the traffic density &t later times:

70.2 Assume there is nearly uniform light traffic on & semi-infinite highway. Sup- pose initially the traffic was exactly uniform p(x, 0) =Po, but the traffic density was prescribed at the entrance, being slightly different from Po, P(o, Po + eg), (> 0, where 0 <eg() < Po: Determine the traffic density &t later times:



Answers

Time of Traffic Fatality In Exercise 51 of the previous section, the probability density function for the number of fatal traffic accidents was found to be
$$S(t)=\frac{1}{91,762}\left(-2.416 t^{3}+90.91 t^{2}-846.8 t+4880\right)$$
where $t$ is the number of hours since midnight on $[0,24]$. Calculate the expected time of day at which a fatal accident will occur. Source: The National Highway Traffic Safety Administration.

Discussion we have or probably didn't density function. And we're going to find sector William, you so, as we know, Victor will you or mean really was doing by new is equal to integration off into a DVD on the Internet. And to be no in this question or interests 0 to 24. So is equal to zero. And they go to 24. No, the video, eh, 40 in the given integration Integration off 24. Can you Here Do you need me? 18 by 91762 into. 0 60 This took you loss 90.9 won t square minus 46.8 loss. Warned of a navy seal DD no. As you can see here, one by 91762 is constant. So, David out said and, uh, moody Play bait. See its jump here. Help me. So here. New but one by 9 70 extremity is constant integration. ALS 0 24 struggled for once. The deed is 24 It was muddy bloody in it, John. Plus 9.91 See you minus 8 46 point square Plus for you know, deep into lady. No digging digression. Do you just root for Hey, integration off! Devious to forests. Do this to five on five. No. 98.91 in division of deep You please do this to four by four miners. 8 46.8 Indeed. Additional T square, ese Tedious to three related by three laws. Integration off two years. It is very divided by two No, our Lord. Limited seal and a problem. Beauties 24 now executed. Upper limit minus lordly me. So you're a mule. 9162 You have 2.4 things. Your upper limit is 20 more. This too. Well, you read it by, right? No. It's 90 point for this board. Four line. It's 46.8. 24 rest too. Three last for about a zero 24. Rest would do it. I do. And our Lord Limited's CEO. So we don't need to write all this part. It was lower limit is already zero. No, my simplify system we get or a mule or begins Victor released starting points to 36. Which is approximate Are yours? Don't midnight. So if you consider midnight that Bradley then after 13 hours soon be one year. And if you consider midnight at three AM then our answers will be for you. Thank you.

So in this question, we have our probability density function of 40 on. Do we have to find our expected value or expected time? So which is new and by music Will do integration off. Yeah, 15 duty. Did he? And, uh, we'll limit is Toby No, Put the value off. A 40 here is equal to zero and busy gold 24 which is given integration. Now put the really work here for tea case 99762 group for 1 60 cube plus nine zero lane square minus 46.18. Last four regulates steel. No, As you can see here, one by 91762 is constant so we can take outside of the integral and multiply D It is here. So your new musical toe one by 91762 Intuition off serial 24. No, your money like d So do you Just 24 year last 909 Dig you my nice. Let's do 6.8 t square. Yours horrible zero t include eady. Don't wreck it. No taking tradition. No international treaties to forties. Do you destroy my You edit by now Integration off Decugis dearest to for directed by four integration off peace rallies. Did you do you write in by to eat and the integration off these t square ready by two. You know, uh, limited. Ciro 24. No executor. Upper limit miners lower limit. So here. Opportunities 24. So here, please put Daisy will do. 24. So here, minus 2.416 24. This truth. Oh, my Devoted my fly plus 9 0.91 liners. 24 rest of formulated by four You're were multiplication sign minus 8 46.8 24 to 3, divided by three. Plus for the blades. Zero 24 mister two divided by two minus. Lower limit is Ciro's where we don't need to write all these things. No, my single. If I well, this thing, then we get starting 0.0 36 hours. Dr Made night. Now, if you consider midnight at well am 10. You and diamond, then your answer. These one B M. Now, if you consider made night at 3 a.m. then oh, our service should be for being okay. Thank you.

Okay, so for this problem are given a density function s a t. And we're told to find the expectation of teeth to find this we're gonna integrate over our whole interval 0 to 24 our density function as a t times t because that's what we're trying to find. The expectations are variable. So instead of free writing out all of this, I'm just gonna write sf t in here. But just know that, uh, this is the function that we're working with, The sense this'll part is a constant. We can have moved out front of our interval when evaluate and the rest of this is all a polynomial times t So all of the powers were just going to go up one and ah, when we evaluate the integral will get something nice and easy to work with. Well, kind of eso we'll get this are in here. Rt will have been to the fore after we multiply by this tea So the interval will give us teach the fifth and then here will have 99.1 1/4 t to the fourth minus 9 64.6 over three Teach the third plus fire. 63 1/2 times t squared and we'll have this whole thing. Ah, scoot over. Just a little bit. Evaluated from zero 2 24 Okay. And so my screen could stop sliding down on me Here, give me just So if we went ahead and plugged in zero into all of these, you would see that all the terms are gonna cancel out. So we only have to worry about plugging in 24 and seeing what we get. And so when we do that, I'm gonna save a bit of time and just plug it into a calculator. Uh, when you do that, we, uh, 12 point 9644 If we round this up to the nearest whole number, we're gonna have 13. So that's the expected T. But our question is actually asking, uh, the time of day that this corresponds to and our time of zero corresponds to midnight, so 12 hours after that would be 12 PM so 13 hours after that will be will be 1 p.m. One. And so that is our final answer.

For this problem. The first thing we need to do is identify what happens to the traffic and 8 a.m. 12 noon and 5 p.m. So we need to have these within the context of that given equation. So you know that T equals zero is 6 a.m. So then 8 a.m. is t equals two, right, cause then teak was one would be seven than eight. This allows us to do some synthetic division to solve for what happens at 8 a.m. For the traffic, right? So then, too, is it divisor? And then we have are coefficients Negative one 25 negative 1 92 for 32 and zero. And this is through the remainder theorem saying that if our T value is to then whatever this value is is what happens at that t value. Okay, It's the first time drop straight down, multiplied by two negative two plus 25 is 23 again multiplied by two 46. We'll continue this pattern, right? Seven Oops. When X equals two or why if you will t value says the T values to the traffic is to 80. But not only is it to 80? It's to 80 and notice that it's positive. So it's greater than zero and zero is the average. So it's to 80. A. Both average. Okay, so then we follow the same pattern with 12. New if eight is too thin. 12 is six. All right, so we do the same thing. No surprises. Pretty straightforward and follow the same procedure since six times 19 is 1 40 14 right, So then our value here is negative to 16 and no negative. So it's below average, and then we don't actually need that negative sign here because we're too. Noting that it's negative by saying that it's below all right, and then make a little room for myself here for our last one, which is 5 p.m. Translating to 11 t equals 11. So once again, same pattern as before. Make sure you keep track of your negative signs so that you don't make any careless mistakes. It's a very that. Then again, that's my bed. It says 14 1 54 and then that equals negative. 38 got ahead of myself. 18. It's than 1 54 right? So then transiting that back to our problem here. It's positive. So it's 1 54 above average, and that answers part ache. Part B, then is to use rational zero serum to solve for the zeros of this equation. Now the rational zeros says that we have to take 4 32 and our leading coefficient, which here to negative one. And it's saying that all of the factors of 4 32 let's call those p and all of the factors of negative one que Then that's saying, with all those options of the form P divided by Q, are possible rational Sears into function so there won't be any zeros outside of this list, But it doesn't guarantee that every zero in this list is a zero up this function. Okay, so beginning, positive or negative one it's possible for 32 is even so positive or negative, too. Rights that I'm going through all of the factors of 4 32 actually has a lot. But when she did notice that I'm not going because like 4 32 actually has huge factors like 1 92 we're not gonna deal with those because think about time. If his T equals zero is 6 a.m. Then t equals 24 is also 6 a.m. There's no reason for us to go that high, so I'm just going up to 23 which, actually 18 is the highest one possible. But then again, we're dealing with time, so we're only gonna deal with positive values because we time goes up, out, down. Okay, so then, considering with the work we just did in part, a two is not a zero and six is not a zero in case we took care of those options. Now we have to do a little bit of tedious work to check these other ones. Okay, so we have one check, if one is zero, and this is the same process we use synthetic division to check. And then for this 1st 1 is gonna become pretty clear that it is not a zero. He's not worried about that one. All right, so then let's try three based on that 66. Okay, so that is also not a zero. All right. Next we have four, and this is, unfortunately a very tedious process. Think of it as guests and check your making an educated guess, and you're just check and see if it's right. So far, we have not been right, but they are still good. Check. Good guesses. Just because they haven't been right doesn't mean they were bad guesses. Necessarily. It just means they weren't quite right. And then here we go. We have a zero. So four is a zero. Okay, So then that means our job became a little easier because we can now use this new polynomial to continue checking, or it's the next I'm gonna try eight, but then you want to just make sure. I mean, I'm sure you do by now, but you definitely want a calculator for this process. Or else you will spin your head in circles around these numbers, and it's a good chance you'll make a careless mistake. All right, so that is not what we wanted. So eight is not one. All right. So, nine. And then again, I'm taking this resulting polynomial instead of starting from scratch. And we came out with the winner nine as a zero. And then this allows us to say OK, the resulting polynomial then is X negative. X squared? No, sorry. Just x to the first plus 12 set that equal to zero and 12 to both sides. I sleep the variable may get X equals 12. So positive 12 with our last serum. And then here's where I'm going to stop because I have three identified here, and t equals zero is also zero The function. The degree is four telling me I have four zeros, so I've identified them all. Okay, so now I'm gonna clear out my workspace cause I've taken care of what I need, and then it can continue moving on to the last step of this problem, which is that we're going to utilize the zeros that we just solved for. Okay, so the zeros we found X equals zero X equals four X equals nine and X equals positive 12. Okay, so then this frees us up to grab Yes. And then this in mind one that get my graph here in order so that it's useful for us. Okay. And then with this, we're also identifying the medics and the men minimum sorry of this function, but we're only making estimates, which is huge. Okay, so then drawing in my two axes and I'm counting by twos here just to allow myself some space, and then it will be a little more readable this way. Okay, so that are zeros. So the origin Adam it go ahead and count these as two. So this is gonna be to I mean, sorry. That's gonna be four. And then so for eight. So this is gonna be nine, and this will be 12. Okay? And then we have our values up here. Where we found to to 80 11 is 1 54 and six is negative. 2 16 Okay, so then, for my y axis, I'm gonna count by 50 since me 5100 1 50 200. So if to is to 80 then it's about right here. So I'm slipped up right there. So 102 100 to 50. So about right here, Okay. And then six is negative. 2 16 So 102 100 right there and then 11 is 1 54 So Ok, so then this allows me to connect and sketch my graph, and that's what we end up with. Okay, so then you want to use your calculator to cut, calculate thes maximum minimums and you'll find you're made. Clear this out. So I have room to write. So you see that at 8 a.m. Were at 2 50 So it's smart to check around that area, putting in pot potential t values. And we'll get about 300 above average again above, just noting that were above zero. And that's gonna be at 7:30 a.m. Which would be a T value of 1.5 and then our minimum we see at six, which is about 2 20 below average. Right? So that's why we're not writing the negative sign because of that below. So it 6 p.m. Oh, that is my bed. That should be 5 p.m. And I've written it wrong in multiple spots. I'm so sorry. The numbers are all right. Do you really? All right. That is six. So excuse me. All mix up. Those are good. Supposed to be six at 5 p.m. Which is a T value. Okay, T value of six at new. And I'm sorry that I got all mixed up, but that does make sense, right, because it's not in rush hour. So basically, this is a big problem of analyzing rush hour versus non rush hour, and we have all the information we need, and we're all done


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