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Suppose, that in town X the number of people P(t), who have heard rumor changes proportion to the number of pcople who have heard the rumor and the number of Assume...

Question

Suppose, that in town X the number of people P(t), who have heard rumor changes proportion to the number of pcople who have heard the rumor and the number of Assume that town X has fixed total population people who have not heard the rumor of N people: (a) Provide differential equation that models hOw' rumor spreads in town X. (6) Draw phase line for your DE assu.ing your constant of proportionality is neg- ative. (c) Draw phase line for Four DE assuming YOUr constant of proportionality is

Suppose, that in town X the number of people P(t), who have heard rumor changes proportion to the number of pcople who have heard the rumor and the number of Assume that town X has fixed total population people who have not heard the rumor of N people: (a) Provide differential equation that models hOw' rumor spreads in town X. (6) Draw phase line for your DE assu.ing your constant of proportionality is neg- ative. (c) Draw phase line for Four DE assuming YOUr constant of proportionality is pos- ilive



Answers

Spread of a Rumor A rumor spreads through a small town. Let $y(t)$ be the fraction of the population that has heard the rumor at time $t$ and assume that the rate at which the rumor spreads is proportional to the product of the fraction $y$ of the population that has heard the rumor and the fraction $1-y$ that has not yet heard the rumor.
$$
\begin{array}{l}{\text { (a) Write down the differential equation satisfied by } y \text { in terms of a }} \\ {\text { proportionality factor } k .} \\ {\text { (b) Find } k \text { (in units of day }^{-1} ), \text { assuming that } 10 \% \text { of the population }} \\ {\text { knows the rumor at } t=0 \text { and } 40 \% \text { knows it at } t=2 \text { days. }} \\ {\text { (c) Using the assumptions of part (b), determine when } 75 \% \text { of the }} \\ {\text { population will know the rumor. }}\end{array}
$$

Hello there. In the following exercise we have the following. So let's suppose that at some time T the population is undermined by a function P of T. Okay. And this function is growing at the rate that is proportional to the population at that specific moment. So we need to determine what's going to be the act possible expression for the growth rate of this function. And here we're going to have as a constant of proportionality. The value of K. Fuck. Okay. So technically we know that the growth rate is determined by the derivative of the function. So that means that the derivative of the function is going to be proportional to the function itself at that specific time. But what this phrase here is saying, okay, and then we have a constant of proportionality. So the idea is a big prime. The T will be equals to this constant K. Times the function be avoided at T. Yeah. And now, so that's the idea. We need to verify that the following function pft equals two A. E. K. T. Satisfied that property. For all the values of A. So we need to observe that this function satisfy this property that we right in the previous part. So the derivative of this function, the derivative of P with respect to T. Will be a K. E. To Katie but there's equals two K. A E. K. T. And at this point it is clear that this part is equal to Okay times P have alighted at T. And this is just saying that P prime of T is equal to K P. t, and these true for all the constants on the reels.

Hello everybody you are going to solve a problem number. Um 48. For first order linear differential equations. Chapter Bernoulli's equation. Okay so at the fairs in the problem what um capital? They're not the population. And or you know the number of people who initiated the rumor? And why equal Y. Of T. Do you not? The number of people who have heard the rumor after 30 days then am minus Y. Is the number of people who have not heard than rumor after two days. And like K. Okay. Do you know the constant of proportionality? Then the differential equations will be D. Y. By D. T. Equal K. Multiply. Why multiply M minus why? Okay. So models the rate, the rate at which the rumor spread for tea. Okay. For t. Wait prison or equal zero. Okay so Y. Of T. Will be equal our capital. Um over are lost M minus or multiply E. To the power negative K. M. T. Okay so and models the function for T graders and vero rewrite this question using equal 5000 and are equal 100. Okay so the question will be whitey. Okay so we can use that are equal 100 And I am equal 5000. Okay so white equal will be equal five 5500 0.0 00 over 100 plus 4900. Eat with about negative five 1000. Okay. D. Okay. And the number of people who have heard the rumor after three days is five 100. So use y. or three equal 5 100 to Find. Kay. Okay so at Y. Three equal 500. Okay so 500 equal 500 on the zero over 100 Plus 4900. He to the ball negative. 5000. Okay multiply three. Okay so by simplifications that equation. Okay so 100 will be equal 49 00. Eat the Power -1. 500. Okay, coma Will be equal one servant. Okay so E to the bar negative. 15,000. Okay. Will be equal Mind over 49 after simplification. Okay so finally negative 15,000 K. Will be equal Lynn. Mind for D. Nine. Bye bye. Taking Lynn for the both sides. Okay so. Okay. Will be equal Lynn Mine over 49. Over 15. Uh 15,000. Oh we have negative. Okay that's value of K. Okay, So for number eight the differential equation will be okay. Will be D. Y. By D. T. Will be equal negative. Len Mine over 14 9 over 15 1000. Okay multiply. Why multiply 5000 negative. Why? Okay. So that the general differential equation models the rate of that spread Of the Rumor. 40. Great to present or equal zero. Okay. And the solution for that differential equation is phone number be. Okay. So Y. Of T. Will be equal 500.0 comma 000 over 100 plus 49 +00. E. To the power de lin 49 39 over 49. All over. Pretty. Okay. That's for T greater than or equal zero for number C. Okay. For numbers the the number of people who have heard the rumor after 10 days. Okay. Okay so why of 8? 13 days will be equal by substituting directly in the French equation. Was the general solution? 5.0 comma 000 over 100 plus 49 +00 E. To the power eight Lynn. Mind over 14 9 Mall over three. So it's almost equal 32 five mine. Okay. Okay that people Okay. Phone number D. Okay for number D. To find how long it will take for half half of the people to hear the remote sold Y. of T. equals 25 Y. Of T. Equal 2050 for T sold 40. So 2,500,000 story equal 500 comma 000 over 100. Last 49 00 E. To the power T Lend mine over 14 9. All over three. Okay so after simplification It will be 100. Okay sorry it will be 49 00. to the power T Lynn Mind over 14 9 over three will be equal 20 will be equal story 100. Okay so E. To the power T Linen 90/49 Over three. It will be equal 1/49. Okay so finally by taking land for both sides. T will be equal three Lynn 1/49. Okay over Lynn Mine over 40 mine. So almost seven. Okay. So it will take about seven days for half of the people to hear the drama. Thanks for watching and see you later.

So they first tell us that they have some model here where why is going to represent the fraction of people that know some rumor and the rate at which this rumor changes or that it spread, which is the same thing is saying D y by D t is going to be proportional to the amount that no, and the amount that don't know. So what all kind of right, Because we still need to kind of write This is some kind of equation. So this is really just like a direct variation problem. So we have d y over DT, and this is going to be equal to or some constant K times? Well, no, the fraction is supposed to be. Why? So we can just write times why and then don't know. Well, if we call this in this why we know that if we add these together, so why plus, in since both of these were supposed to be fractions, some up to 100% or in this case, just what so to solve for what it is going to be, we just attract Why over so that, says Innes, even toe one minus y So that's what we're going to get right here. So one minus y. So we end up with this expression here. So this is going to be part A for us getting everything set up, Um, and probably off on the side. We should just say that K eyes just some riel number, So just some constant. Now, the next thing they want us to do is to solve this differential creator. So let's go ahead and do that so we can go ahead and get all of our wives on one side and all parties on the other. So I'm gonna divide by why one mice y and then multiply by DT. Somebody will do part, be here. So if we do that, we'd end up with d Y over y times. One minus y is equal to K d. T. Now to go ahead and integrate each side. Now, before we can integrate what we have on the left side will have to do a partial fraction decomposition. So I kind of do that off on the side here. So we want one over Why? Times one minus y to be equal to a over y plus 1/1 minus y. Because remember, not one B or a B or just some constants. And then if we can get it into this form, knows we know how to integrate a over Why? Because that's natural log of the absolute value of why it and then over here for one minus y over one that's going to be the negative natural log of absolute value of that. So we go ahead and use that there. Um, yes, I remember at this point, we clear denominators, so I'm gonna multiply by y and one minus y. So over on the left, we just have one is equal to and then it's a times. Well, if I multiply by why that white have sought one minus y. So I'm just left with one minus y and then be the one minus y scans out and we're just left with wise would be B y. So this is going to be our equation. We need to salt, and in this case we can start by saying, Let's do why is equal to zero because that will cancel or be terms. So that would give us one is equal to, uh, a one minus Seriously, that's just one. And then now just be plus syrup. So that tells us a Is he gonna one? So let's come back up here and erase this A and put one there. And now we can do Why is he the one? Because notice that cancels out the terms would be one is equal to, um, zero plus B times one. And so then that says B is also equal one. So a and B are just going to be one. And if you were to, like, cross, multiply or add these, then you'll see we'll get the exact same equation. So that will say that checks out. So let's go ahead and rewrite that over here. So this is going to be the integral one over y plus 1/1 minus y. Do you? Why? And then on the right here, wolf, we integrate K. Remember, K is just some riel number or just a constant. So if we integrate, that should give us Katie, Then we'll have plus some other constant see. But this is our integration constant here right now, actually. Let me close that off. Right. So now we can integrate one of the white because this natural log of the absolute value of life and then if we were to integrate 1/1 minus y, that's going to be minus natural log of the absolute value of one minus y and remember, is going to be negative out here. Because if you were to do, like a use substitution, um, we do. I'll just write this right here. So you is even a one minus y. Take the derivative. That would tell us D u is equal to negative d y. So that's why that negative kind of comes out. Um, and then this is equal to Katie, plus C. So now our whole goal is to try to solve for why? So let's go ahead and combine these in tow one logarithms Because, remember, when we're subtracting logs, we can rewrite this as one log by dividing what's on the inside and then to get rid of the log, we can exponentially eight each side. So then this is going to be, uh, absolute value of lie over one mice Wife is equal to e to the K T plus C. But remember this we could actually rewrite to be e to the k t times e to the C and then e to the C. It's just another constant. So let's call that another seat. So remember this c and this C or different seeds? But normally, Moamer solving differential equations were kind of sloppy. And we just keep using, see over and over again, right? But now we can use that. And so let's just write that also, it's absolute value of why one mine Squire is equal to C E to the Katie. And now, at this point, we can actually drop the absolute value, because before this quantity here could have been negative, um, and e to the Katie policy is always positive, so we couldn't drop it. But here we can, because notice see can just be some negative numbers. So in this case, we can actually drop that with no issue. Yeah, eso Now what all do is multiple each side by one minus y. And doing that will give why is equal to. Maybe I'll write this on the outside here, so do one minus y multiplied. So that would be C E to the Katie minus c y e to the Katie and I'm going to add this over. So it's gonna be why. Plus, see why e to the Katie is able to ce to the Katie. And I do this here because we want to collect the wise on the same side. So we get factor it out. So now we go ahead, factor out the wise. That's why one plus c e to the Katie is able to see heat Katie. Then we get divide that over. So that says, why is equal to C E to the Katie over one plus ce to the Katie. Now, at this point, we could just leave it like this. This is fine for the most general solution. But notice that we can divide the top and bottom by e to the K t. And that will only leave us with, like, one thing to input. So one over or Beria instead of one over e to the Katie are multiplied by E to the negative, Katie, because that's the same thing is divided by this. Okay, And then once we do that, we get why is eager to sort of BC over e to the negative. Katie here and then plus see like that. And now we just need to figure out Well, what are we going to plug in for? See here? Because we want a more general solution. Well, if we kind of go back, we really don't know much from the problem. Other than just why is the fraction that? No. So what we can kind of do is just say, Well, what if at time t 0 to 0, we just have where? Why is ableto why not? Which is the initial percent that no, or I should probably say fraction that No. And then we could just go ahead and solve for C this way. But notice if we were to plug that into here is going to be, like, really complicated. So if we think about it, we could actually go back to the last time that we made any kind of, like, weird changes to see and that actually occurred right here. So let's actually a race everything that we had here. So this, see, remember, is the same see down here because even when we like multiplied, we didn't like, absorb it in or anything like that so we can apply what we just said. That t is equal to zero. Why is it so? Why not and then solve for C here? And if we end up doing that, that would give us why not over one minus. Why not? Is equal to see times e to the zero and eat. The zero is just one. So that means, see, is why not over one minus? Why not? So let's come down here and plug that in. So, um, we said c is able to Why not over one minus? Why not? And if we were to plug that into that equation, that would give us why is equal to why not over one minus. Why not all over e to the k t plus, why not over one minus? Why not? And again, I really don't kind of like this one minus. Why not being there? So you could technically leave it like this. This is a valid way, but similar to what we did before, where we multiplied by hope This is e to the negative k. Do I write into negative Kathy idea eso similar to what we did when we multiplied by each negative K to kind of simplify it further. I'm going to multiply the top of the bottom by one minus. Why not? So this isn't really needed. It just makes it a little bit prettier, toe. Look at. And now that gives us why is equal to, uh why not over so one minus. Why not get distribute over here? So it's one minus. Why not e to the negative, Katie, And then Plus, why not? And so this is going to be our solution. And again, remember, why not is equal to percent or fraction fraction that No. Okay, so that was the end of part B. Now for part C, um, they tell us what kind of write this down. So for part C, there is a population of 1000 people in this town and at 8 a.m. 80 No. So 80 people know about this rumor, and then they say at noon, half no. So from this day wants to figure out. So when will 90% No. Right. And these two things here are essentially, like our initial conditions. So this is really telling us that he zero because this is going to be our starting time. because it's starting at 8 a.m. And then they say 80. No, but remember, it's not just going to be 80 because why is a fraction? So this is actually telling us Why not? Is going to be 80 over 1000, um, or 0.8 So if we actually come back up here till what we got, we could just replace Why not with 0.8 So actually, I'll do that. I'll be a little bit lazy and scoop this down, and then I'll replace this with 0.8 This the 0.8 and this with 0.8 All right, so let's get rid of all that now. So that was using that first conditioner. Now for the second one, it tells us at noon, half no well, noon. If this is t zero. Well, this is supposed to be four hours later, since noon is at 12. So that's really saying when T is equal to four, why is going to be equal 2.5 or one half so 0.5, since half of the population is going to know, and then we can use this because, remember, we still don't know what K is. So even if we were to try to, like, set this equal to 90 we'd have to figure out what K is before we can proceed. So let's go ahead and plug in 900.5. So there's going to be 0.5 is equal to 0.8 over and then one minus 0.8 is 0.92 e to the negative. 80 plus 0.8 And now we just need to solve for this. So first, I'm going to divide over by 0.8 So 0.5 divided by 0.8 So that gives 6.25 is equal to 1/0 0.92 e to the negative. Katie lost 0.8 We can reciprocate each side. So then one divided by 6.25 is going to be 0.16 is equal to, and then we flip over here. Well, the new emerges becomes the denominator. So 0.92 eat the negative. Katie plus 0.8 Now we can subtract 0.8 over and divide by 0.92 So that would be minus 0.8. Open up 0.8 0.81 point six finest points or eight psychosis points or eight, and then divide by 80.92 So that is going to give us something around zero. Actually, let me just see this as a fraction. So if we divide and all that that would give us to over 23 is able to eat the negative K t. And now we can just take natural log on each side. Natural log, natural log, natural log in the accounts out over here is just negative, Katie. And actually, I guess I forgot to do this. But this should have been four. Because remember, we said our time waas for it, this case. So we should actually go back and replace all these teas here with force, because otherwise we have two variables that we really can't solve. Great Here. Now we have that. And then we just need to divide by negative for over. So that tells us K is equal to negative 1/4 match log of to over 23 which this is approximately. Let's see eso natural like that isn't about negative. 0.24 and then divide by negative four. So that's saying this is about 0.610586 And I'm just gonna leave. That is K. Um, and then we'll just plug this number in because I don't wanna have to, like, plug that into this equation up here. Okay? But now if we want to figure out well, where is it? Population? Half. We're going to take this same equation that we had right here. But instead of doing one half What president? That's right. Yeah. So instead of doing E, just lean on that. So instead of plugging in half, we wanna know 90%. So in this case, we'd have 0.9 over here, and they were just gonna repeat the exact same procedure that we did for the last one. Yeah, and so then remember, we just need to keep in mind that k is this 0.61 number, or I guess, that negative national log that we have there. Okay, So let's go ahead and just go through and solve for everything and then and we'll plug all that in so same steps. So we divide over. So 0.9, divided by 0.8 So that gives us 11.25 is eager to 1/0 0.92 e to the negative. Katie plus 0.8 we can reciprocate each side. So if we divide 1/11 0.25 that gives 4/45 and then over here 0.92 e to the negative. Katie plus 0.8 So we subtract 0.8 Divide by 0.92 and then that gives to over 207 is even to e to the negative. Katie. And then again, we're going to take natural log on each side. Natural log, natural log those counts out negative, Katie. And then we divide by negative K this time as opposed to t. So that says T is equal to, um negative one over k natural log of tea. And let's see, Let's come back up here and plug in What? Caisse, I'll just kind of suit this down. Okay, Is that? And if we were to plug this in Well, this is supposed to be one over this now, so one over all of that. So it really just reciprocates the 1/4. So this is going to be equal to on the negatives Will actually cancel out with each other as well. So this would be four times natural log of why did I put t here? This was 207 to over two or seven to over 27 and then natural log of to over 23 years. So this is the exact time that it occurs, but I mean or how long after it occurs. But I mean, to us, that really doesn't mean much. So let's figure out what that would be an actual time. So we have four natural log of two, divided by 207 and then divided by natural log of two divided by 23. And that gives something around 7.598 hours. So what this says is 7.598 hours after the fact is when we would reach 90%. But we started at eight. So actually, let's first convert this into hours and minutes. So this is seven hours. And then to convert that into minutes, we would just multiply the decimal part by 60 So if we subtract seven from the number we just got and then multiply 0.598 by 60 that would give us, um, 35.9. So I'll just round that up to 36. It would be 36 minutes. So we'll take this and then let's come back up here. So we said we start at eight, and then we're adding seven hours and 36 minutes, so that means 90% will know. So it be 8 a.m. You should let me write this off on the side after I block this off. So if t is equal to zero is 8 a.m. and then we have t is equal to seven hours, 36 minutes. That means this new time is going to be s 07 hours after eight is going to be three and then 36. So it's just 36 minutes. So would be 3. 36 Pete. So this is the time that 90% of the population will know about this rumor

So first we want to know when half the population. Here's the rumor. Eso to do this, we're going to essentially start with saying that this is going to be one half, so we'll have one half being equal to this whole thing right here. Then we can flip the fraction and we get that two is equal dress aan den. We'll get what's attract one and see that one is equal to that. We can divide by a so we have one over A is equal to this, Then we'll take the natural log of both sides. When we take the natural log of both sides, we see, yeah, that will end up with That's a negative, Katie. And then we can divide by a negative k on. But we'll end up getting is that t is equal to the natural log of one over a divided by a negative K. So with that, we Well, now I want to determine when the spread of the, uh, rumor is at its greatest. So it doesn't matter what we choose for this value doesn't matter Caisse high or what we mean we can just set it equal to one and then analyzed the graph. So looking specifically at this right here, we see that the rate is at its greatest, um, approximately right here. Um, and in general, if we were to look at the derivative, we see that the rate is greatest at the natural log of a divided by. Okay. Lastly, we, um, want Thio sketch the graph of Pete. So in this case, we have a f of tea. But this would be our graph right here and again. We can change the values of A and K, but this is ultimately what our graph is gonna look like.


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