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In 4 small town of population 5000 the rate of growth of an epidemic (rate of change of the number of infected persons) is jointly proportional to the number of peo...

Question

In 4 small town of population 5000 the rate of growth of an epidemic (rate of change of the number of infected persons) is jointly proportional to the number of people infected and the number of peo ple not infected' When 100 people are infecterl, the epidemic is growing at the rate of 9 people per day:(a) Formulate model expressing the rate growth of the epidemic as funetion of number of people infected_(6) How fast is the epiclemic growing when 200) people are inlected?(c) Using a graphin

In 4 small town of population 5000 the rate of growth of an epidemic (rate of change of the number of infected persons) is jointly proportional to the number of people infected and the number of peo ple not infected' When 100 people are infecterl, the epidemic is growing at the rate of 9 people per day: (a) Formulate model expressing the rate growth of the epidemic as funetion of number of people infected_ (6) How fast is the epiclemic growing when 200) people are inlected? (c) Using a graphing tool_ (erit' how MaDY peeple are inlected when the rale epicletic is HAXMm



Answers

Spread of a Disease The rate $r$ at which a disease spreads in a population of size $P$ is jointly proportional to the number $x$ of infected people and the number $P-x$ who are not infected. An infection erupts in a small town that has population $P=5000$ .
(a) Write an equation that expresses $r$ as a function of $x .$ Compare the rate of spread of this infection when 10 people are infected to the rate of spread when 1000 people are infected. Which rate is larger? By what factor?
(c) Calculate the rate of spread when the entire population is infected. Why does this answer make intuitive sense?

So we know the rate of a certain spread of a disease is jointly proportional. So it's going to be a to direct proportions to the number of people who are infected times the number of people who are not infected. And they told us to assume that the population they started out with being just being P and we're to assume that the population is 5000. So there's our equation. So there's some cake concept we don't know. We're never going to find it. The number of people who are infected and the number of people who aren't infected assuming that the population is 5000 people. So that was part a part B. We want to find what happens if the number of people being infected is 10 versus if the number of people being affected infected as 1000. How do those rates compare? And so we would have 10 here and then we would have 5000 minus 10 is going to be 4990 and so this value becomes 49,900 K. And if there are 1000 people, we'd have 1000 people here for X and then putting 1000 in here. This would be 4000. And so if we take that, we'll have four with three zeros and three more zeros. So it's going to be four million mhm times K. So again, we don't know exactly what the rate of infection is. We just know that this one is much larger, so this one is much larger. And by what factor? If we divide these two values and take the four million K and divide it by the 49,900 K, we're going to find that the factor is approximately 80 80 times more Infection Higher infection rate now, Part C says Now what happens to the infection rate If everybody is infected? Well, this would be 5000. Everybody's been infected, and then we'd have 5000 minus 5000. That would be zero. So the the rate of infection would be zero. And does that intuitively makes sense? Yeah, it does. Everybody has the disease. So now the rate of being infected we don't have any infection of new people because everybody has it so intuitively it does make sense

Okay, This is the question Number 54 here. That is the right off Ingres of infection. Your baby is depopulation on ecstasy. Infected people. So he'll rate on Jon Lieber was no. During forget People X and the people who are not infected. So we can write your B minus X in the grand question, your we herpes equal to buy them. So our reports No thanks. My tongue then minus six Now to remove the problems. Now, the reason we need to multiply constant case of your artistic or two x 5000 minus six, right from this aggression, begin right to your own one by r two. Is it going to explain my ex to monitor bye bye. 5000 miners expert like house them minus next to right. So in the first case here, X one is equal to 10. That is even more off inflicted people. We stand and your ex to that east. Each one tells them, right, put more deserve you and disintegration. But here one by r two Is it one of your X minus 10? My one told them, Monty like like 5000 minus 10. Do you want it? But flight times in minus one. So here, for one by our to is a void of your one by hundreds. Multiply by 5000 minus 10 that we see for mind. And Jubal, do you read it by four tons of you're on one directed by our king is equal to me. Stored up on 90 upon your 4000 multi by my 100 eighty's, 100 see was evil. And you So we can right here doing my arm one is equal to for lack the backbone of our land. Zero on to buy Our one is a quarto 80.16 So we can say that the rent X is equal to 1000. That is, the number of infected people are want of them. Then the rate is higher by 80 point once expect Er right now we need to calculate the right. Oh, spread when the entire population is infected. So we can right here. Excuse. It would be right. Put the venue effects in this aggression. If your car is equal to Kmart deployed by X into you B minus the vehicles here X. Is it going to be of your aren't is acquittal gigs, Marty. Glad as you so are easy. Serial. Right? So Begin said that there will be no infection spread when all people are infected. What? Thank you.

So in the town, the number of people infected by a disease at the time he is given by NT is equal to 3500 divided by one plus 19.9 in tow Eaters to the part minus 0.60. The first part of the question is how many are initially infected with the disease. That means I have to put easy called zero. So I'll write, and zero is equal to 3500 divided way one plus here. If I put the oh, this will be 19.9. So this value on simplification comes out to be 1 67 point 46 So I consider awfully 1 68 people are infected at initial period. If in the V option, they're asking for a number of people infected after two days, five days, eight days, 12 days and 16 days. So I have to find this and off to that. When Steve Value, I'm going to put too what is going to happen? 35 Double zero. Divided by one plus 19.9 days to the father, minus 0.6 multiplied by two. So this value on simplification it comes out to be 500.44 Hopefully we can tell 501 people next. Partick, They're asking me off five. That means instead of toe put side. So when I put that five over here, I'll have this value equals two 17 favorite. Find one toe next and off it. That went instead of me. I'm supposed to put it so here I am going to have 3000 7.46 number of people next. But I have to put in Well, so here I can see until off 8 12 and I have done this 34 for it. 0.76 next and 16. So instead of 12 I'll have to put 16. So this value gives me 34 95 point toe nine people so I can see that the number of people in factor is going on decreasing. Okay, It's continuously decreasing. So now the sea barred. We have to see using this model. Can you say whether alter develop out 3500 people will be ever be infected? All right. So I can see that as the time keeps on progressing, I have seen that n zero is equal to 1 67 0.46 Andi over the period of time. If I put an infinity over in finite period of time, if I consider what is going to happen, it will be 3500 divided by one plus 19.9 in tow. Eaters to the par minus Infinity it'll become. So here's Tolliver minus Infinity is basically zero. So then, in that case, 3500 divided by one plus zero so that when he was going to be 3500. So I can see that over in finite period of time, all 3500 people can get infected. Okay, But that practically requires in finite amount of time. So I love to stay here very clearly. All 3500 people will be infected over and in finance petered off. So basically, if I consider the time in trouble, Toby Very large cocaine shirt off 16 days or something. If I consider one year or something like 3 65 days or 405 100 days, then definitely this where you will be approaching toe 3500 but in finite time is just the assumption, because it will just keep getting closer and closer to 3500. But since then, finally we cannot count. So that is why 3500 also will not be ever achieved for practical reasons but mathematical reasons, it is possible so that it thank you.

And this problem. We are analyzing the behavior and the different conditions of an exponential function. And we're going to use three exponential function to predict different values at specific points. And then we're going to be introduced to the rudimentary elements of what a limit is. So we're given the function. End of T equals 3500 over one plus 19.9 times e raced negative 0.6 t. So the first thing that we need to do is graph it. So you could just plug this into a graphing utility, a graphing calculator, and you would find that our graph has the shape. It increases exponentially. And then it's going toe level off for Part B. We were asked what's going to be the initial value of this function so you could do this one of two ways. You can either look at the graph and trace the points, and you'll find it that way, or you can do it. Algebraic Lee. So this is when our initial value So this is going to be when T equals zero so you can plug in zero for tea into our function and you'll get the initial value so end of zero is going to be equal to 3500 over one, plus 19.9 times e raised to negative 0.6 times zero. You can plug that into a calculator, and we'll find that the initial value is 167 people. And now for C. We're basically doing the same processes we just did. We're just now asked what's going to be the value of our function at specific points. So this first one end of to do the exact same thing you did with zero has plug in to instead of zero. We would get 3500 over one plus 19.9 times e raised to negative 0.6 times two, and we'll get 500.5 now for these other values. Do the exact same thing. Just plug in that number for tea. So when you do that for end of five would get 1758.12 for Innovate. We would get 3000 and 7.46 for end of 12. We would get 3448.76 for N F 16 we will get 3495.3. And now we're asked what's going to happen with this function as T approaches zero Now, the question isn't phrased like that. The question is asking us, Are all 3500 people going to be, um, are all these? Are all of these 3500 people going to contact this disease? That's a better way to put it. Well, what does this mean mathematically? Well, what's going to happen when our value t approaches infinity, all of our population, while the denominator is going to approach one as he approaches infinity? How did you get that Will plug in infinity 40 and see what happens to that exponential term. So what does this mean? This means that our function is going to get very, very close to 3500, but it's never actually going to reach it. So not all people would contact that disease. So I hope that this problem help to understand how we can interpret the behavior of an exponential function and then start thinking about the concept of a limit where a variable approaches a specific value. In this case, when he approaches infinity


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