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The population of a town is growing at a rate givenby dPdt=144−16t13dPdt=144−16t13 people per year. Find afunction to describe the population t years fr...

Question

The population of a town is growing at a rate givenby dPdt=144−16t13dPdt=144−16t13 people per year. Find afunction to describe the population t years from now ifthe present population is 85008500 people

The population of a town is growing at a rate given by dPdt=144−16t13dPdt=144−16t13 people per year. Find a function to describe the population t years from now if the present population is 85008500 people



Answers

The 2010 census $^{12}$ determined the population of the US was 308.75 million on April 1,2010 . If the population was increasing exponentially at a rate of 2.85 million per year on that date, find a formula for the population as a function of time, $t$, in years since that date.

Guess who asked. If I m t. That's reaches the population size of one millions. That's 25611 for each is over 25 Is he going to want one okay itself? It is. This is 1,000,000 23123 over 25611 for Take the island on both sides. Divide by 0.2 fine in this iniquity, but it isn't exactly later going up. One joke. No, no, no, no. By 261 before divided by point right, his reading's approximately t. It's five years.

So in this model of a population here we have an initial population of 500 we're growing at a rate of 24 per year. Um, from 2010. So our population, as a function of t is gonna be We'll do our rate times, um, our rate times every year. So 24 t and then we'll add our initial population plus 500. So this is going to be our function of the population and then now to graft that well, that it looks like is that will start. So we have t here and then pee here. And then now we're going to start at 500 and then we're just gonna go up by 24. Okay, so this is 500 here now to predict the model of the population in 25 2025 2025 is 15 years, right? So, um, 15 years after 2010. So we're gonna plug in 15 so p of 15 is going to be 24 times 15 plus 500 24 times 15. That's equal to 360. So we have is this is equal to 360 plus 500. So our population in 2025 is going to be equal to 860

Let's talk about question about 57. Uh In this case it talks about the population growth. So it's a world problem says that the population p of a city grows exponentially according to the function. So the function is given pd north probably the population. And that's a function of T. Which is time. So P. T. Is 8500 times 1.1 race to T. Where The ranges from 0 to 8. Uh T. Is measured in years. So 0 to 8 years. We have to find the population at the time to equal zero, and also at the time article. So this is a pretty straightforward one. The population has already given to us in the form of 8500 times 1.1 raised to so at equal to zero. What we have to do is we just have to replace T by zero. So we need to find the value of P zero which is 8500 times 1.1 race, 20 and 1.1 raised to zero is nothing but one. So the value of P zero comes out as 8005 years. This is the first uh bit of this particular part. For the next one. We got to find the value at table 22 So likewise, we gotta find P. Two. It means that we have to replace uh t by two. So that's gonna be 1.1 square. So uh we need to find 1.1. We're going to take the help of calculator here, 1.1 square times 8 85 100. So that's going to come out as 10285 This is the value of population at our time equal to. So this is the part A. Let's talk about the part B. The part B is asking us the ear the nearest year when the population will reach 15,000. So in other words, we are given the value of PT as 15,000 and we need to find out what is the time when this happened. So what we're gonna do is we're gonna substitute the value of PT in the situation. It's gonna look like 15,000 as you can do. 8500 times 1.1 raised today. So this three, uh Dividing both sides by hundreds or two Zeros get cancer. So we are left at 150 years ago to 85 times 1.1 race to teach if we divide both sides by 85. So we have 1 50/85 is equal to 1.1 race to T. This means that 1.1 race two T is equal to 1 50 or 85. That's another way to look for to rewrite this. And since we have to win the value of t shirts, a good idea, If you take a log both sides, a log of 1.1 race two T. Is equal to log off. Mhm. 1 50/85. So as for the rule of the law algorithms, a lot of areas to be so one which is in power just drops down. So we have a we have flexibility in this way, three times log of 1.1 is going to be able to log of 1 50/85. And if we divide both sides by log of 1.1, the value of t comes out as log of 150 over 85 over log of 1.1. So this is where we need to take the help of calculator. And if we do that, the value of T. Consort as religious one second. Let me grab my calculator here. So 1 50 or 85 log of this value. And it should be divided by the log of. They should be divided by log of 1.21. uh so this is the value of T. When this happens, that's going to be five nine for 96. But we need to round to the nearest year. So to the nearest year, definitely 5.96 can be longer too uh Six. Or the value of T. For this is six. When the population uh for this particular function becomes 15,000. Thank you.

So we're showing that the population of the region Um is growing exponentially. There are 40 million people in 1990. Which is when t. equals zero and 56 million people in 2000. So we want to find an expression for the population at any time. T. So we know that our population models can be T. Equals do not. Me Too. The rt the issue is we need to find the rate of population growth because we know that the initial population Is going to be 40 million. So we're just gonna write this as 40. We'll talk about the population in millions And then we reaches 56 million. And that's when T. is 2000. So that's 10 years later. Mhm. So since that's the case, we see that we can divide 40 on both sides, Take the natural log on both sides giving us That's right here and then we see that our is going to equal 1/10. Yes, we end up getting that as our rate and we'll call this actually RK constant. So now we have P equals peanut, which is 40 E. To the K. T. So with that we now have our model for the exponential growth of the population.


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