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Exercice Uses the population growth model:The bat population 32 vesrs_certaln Mldwestern county230,000 2012 _ and the observed doubling tme for the Popu aton (a) Fi...

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Exercice Uses the population growth model:The bat population 32 vesrs_certaln Mldwestern county230,000 2012 _ and the observed doubling tme for the Popu aton (a) Find an exponential model n(t) RozUa for the population ycars after 2012.n(c)230,000(2)(6) Find an exponential model n(t} Nog" Tor the poqulalian Ydny Jitcr 2012 (Round Your value placcs )four declman(t}c) Sketch _ graph 0f the populatlon &t time

exercice Uses the population growth model: The bat population 32 vesrs_ certaln Mldwestern county 230,000 2012 _ and the observed doubling tme for the Popu aton (a) Find an exponential model n(t) RozUa for the population ycars after 2012. n(c) 230,000(2) (6) Find an exponential model n(t} Nog" Tor the poqulalian Ydny Jitcr 2012 (Round Your value placcs ) four declma n(t} c) Sketch _ graph 0f the populatlon &t time



Answers

These exercises use the population growth model. Bat Population The bat population in a certain Midwestern county was $350,000$ in 2012 , and the observed doubling time for the population is 25 years. (a) Find an exponential model $n(t)=n_{0} 2^{\prime \prime}$ for the popultion $t$ years after 2012 . (b) Find an exponential model $n(t)=n_{0} e^{n t}$ for the population $t$ years after 2012 . (c) Sketch a graph of the population at time $t .$ (d) Estimate how long it takes the population to reach 2 million.

Back population in year 2012 is 350,000. So consider this as initial population and north on the blink time is 25 years. So a is 25 years. What we can say is 25 now in part, even need to find exponential mortal per population that is NT equals and nor two days to party by a We know the value off and north and a and after substituting these values, the Gator population model it is this so named, he says, a question number one now in part B, we again need to find an explanation mortal for our population in this form which is n p equals and north into it is two part arty. They're artists. I really do growth rate related growth Read now, Firstly, to find this exponential mortal, we need to find the value of our related growth ring that is our so to find our people first, fine. The population so firstly find the population in years 25 years, so four p equals certifying. That is, after 25 years off 2012 our population will be using equation one. The values equals 25 in a question Number one and get and P equals 350,000 into to raise to power 25/25 since they put the value off Big was 25 here and after solving this be good. The population off bags after 25 years off 2012 is 700,000. Now we will use these values to find our related growth Read So what These values in equation number two said this. Therefore, when he is 25 our population is 700,000. So using equation number two which is this our population and 25 ik was and north into eaters to part our into 25 as thes 25. So this becomes 700,000 equals 350,000 into eaters too far. 25 dividing throughout by 350,000 Bigger two equals eaters to Barbara. And if I all And for this we can ride this as I lento equals 25 dividing throughout by 25. We get our value off our bitches. Ellen do over 25 now solving this using calculator The value of r 0.2773 So this is a related growth rate on after putting this stallion, Eric Exponential mortal for population we get and P equals and not into it is to part our which is 0.2773 into OT and since the value of and nor is already known so we will put this here. Therefore, this is an exponential, more doleful population. Namely, says the question number three now in parts in part C, we need deplored the graph off our exponential mortal function. So for some values off B, we will find the corresponding values off empty. That is, for some time after 2012 we will find the population in that year. So initially our population was 350,000 and after 25 years they have calculated that our population became double that this 700,000. Now we will use these two points to upload that exponential function. So initial population is 350,000 and after 25 years, our population becomes 700 1000. Can I think be used to beget a graph like this since it is an explanation function so we can make this at this. Now this is a graph forgiven exponential function. There are XX is taken as population on via taxes on Sorry, where are y? It's is is taken as population over time on acceptance is taken as time in years. Now the sole part B we need to find that Vienna population will reach that is the owner population and we will reach two million No using equation number three. Here we will put the value off and be here and sold for P. Therefore, two million equals 350,000 into eaters to pile the low 0.2773 into P. Dividing throughout by 350,000 we get 5.71 Port three equals it is to par 0.277 three in duty now taking long. Both sides make it Ellen 5.7143 equals 0.2773 And don't be now dividing throughout by 0.2773 big. Now we can solve this valley off long by using calculator and law 5.714 threes, 1.7 poll to 97 and dividing the whole equation by 0.2773 So this is a value that we get. We're solving this 40. We get 62.85 Therefore, our population will reach two million off 63 years after 2012. So are bad. Population will reach 2,000,063 years after 2012.

This problem. We're dealing with the population of a bad egg. We need to find the equation for modeling the population using the doubling time. So for doing so part, eh? The equation is an A T is equal to and not times two to the who's it's gonna be. And not times two to the tea over a power. Now, we were giving her a It is 25 years were given her and not in his 350,000 years. So we can just rewrite this equation as N of t is 350 1000 comma right there Times two to the tea over A, which is t over 25. Now for part B, we have to use the relative growth rate equation, which is an a t this and not times e to the Artie we know and not so we can write this as a robbery or 350,000 times E to the r. D. We don't know our and tea is just an arbitrary of value that we plug in. So what we need to do is somehow cancel out t so we can solve for our and the way we're gonna do that is by setting this equation and this equation equal to each other, since they are both equal to end a teat. And when we do that, we get 350 1000 times two to the T over 25 is equal to 350 1000 times e to the rt right off the bat. 350 thousands cancel and we're left with two times. T over 25 is equal to E T R. T. Now we take the natural log of both sides. When we do that, we can take the exponents here and pull it to the outside of the natural log. So we're left with T over 25 times the natural log of two natural. I've e is just one. So we're left with our tea here. Now our teas cancel up and we're left with our is equal to the natural log to over 25. So, in other words, our is equal to about zero point 0.277 now part See, we have to graph the hopes we should actually write this equation now. Um, so this rewrite as 350 1000 times E to the 0.27 Maybe an extra seven here, seven t good. And now, for part C, we have to graft this equation. So what's your draw? Our first and second quadrant, and we're just gonna draw a basic sketch of what it will look like. So here we have our 0.0 comma, 350 1000 because that is the end, not valued. And we're just gonna draw a very approximate graph of the exponential growth. So something like this and we're good there. Now, for Part D, we have to plug in 2,000,000 in place of end of tea here so we can find out what time the population equals. 2,000,000. So this is going to be I'm just gonna write. This is two mil is equal to 350 1000 times E to the 0.2 7/7 point 0 to 7 70 and from here, divide both sides by 350,000 and you get 5.714 is equal to e to the 0.2 7 70 Now you take the natural log of both sides pulled this exponents outside the natural log. So you're left with The natural log of 5.714 is equal to 0.27 70. Now you divide both sides by 0.277 and you get T is equal to about 63 years. So the population will reach 2,000,000 at about 63 years and we're done.

So we need to find a model, and we know that the number of bats in the year 2000 and nine was supposed to be 350,000, and we wanna find him. I am. We know it's doubling, remember are doubling every 25 years, and we want to write a few models for this. But the catch is also that we need t to be the time after 2006, which this question would be far easier if that were 2006, rather than not 2009. So we are gonna have to do a little bit of figuring. So if we started out, let's just do a calculation to figure out when we look at this doubling model. And again, there are a few algebraic ways we could work with this. But let's just let's just look at this. We want to write that first one as a model that looks like this. So we want to do with that doubling idea, so we know if time were after 2009, then we would start with 350,000 bats and we're having it double and we would be having it double every 25 years. And that would be the model if we were letting t stand for time after 2009. But we want it to be time after 2006. So one thing we can do is we can figure out this again is the model if t is in time off 2009. And so what we can do is say Okay, well, I just want to figure out how Maney bats were there in 2006. So we want to enter a time into this model up negative three because we want to go back three years. So let's figure out what that ISS and again, there's more than one way to do this. But this is just one way. So if I type in negative 34 time and I'll put that in my calculator, haven't tell yet. 350 boom, boom boom 1000 then times to to the power of and I have negative three divided by 25 make sure all of that is in the exponents position and we find out that that means that in 2006 there were 322,000 and 65. I'll say 66 bats all round up. So when we write our model with T being in terms of time after 2006, my model will actually end up looking like this. 322,000 65 Loco 66. And then they're doubling and they're doubling every 25 years. So that would be my model with the way they wrote it. Now, if I didn't have to have and time is time after 2006, if I didn't have to start with that time, I would have written it like this and had time be after 2009. But we weren't allowed to do that. So now on part B, we want to change it so that the model is written in terms of and subzero and A to the rt. So I look at this model and again there are a number of ways of doing it. But I know for sure that my model is gonna end up being 322,000 66 baths and e to the and then will be a T. I just have to figure out what that our value is. And if I look up here, my models almost look identical. Except for this one. Has the base being to to the 1/25 times t and this model has the base being R E to the r times t. So that means I need these two bases to be equal. And I need this thio equal this So I need to find when is to to the one 25th power equal to e to the r. And I can do that a couple different ways. I could take the natural log of both sides and then I can use my log of a power and I have or I could just hit this on my calculator. But on this side would use log of a power are comes down. We have our times, Elena B. And we know Elena B is one. So I get my our value is just this whatever this natural log of to to the one 25th or I could put the one 25th in front, But we'll just hit that natural log of two to the power of one divided by 25 and then we'll with that over closer parentheses. E. And I get that. That decimal comes out to be 0.277 and it continues on its like God to six so I can write my model. And again, this is time after 2006. So I have 322,000 66 bats e to the 660.2 and I'm just gonna say 77 t now we want if you put this into your graph for I'm not gonna show the graph of this, but we we know that we would have the initial amount would have the initial amount would be graphed here, and then it would be doubling every 25 years. So you'd have to have if this were like around 300,000. You know, after 25 years, you're gonna roughly have 600,000 baths, etcetera. But it is gonna be exponential growth. And I'll leave that for you to graph on your calculator. Just get the appropriate window. Then we wanna estimate when this is gonna be two million and instead of estimating, let's just use our model. Let's use the top one. No, and I have that. The 322,000 66 bats, times two to the T over 25. I want to know when that's going to equal two million. And so we know who would start out by dividing, dividing both sides bed at 322,000 66 bats of the initial number of bats in 2006. And that's gonna cancel that. And then we would take the log of both sides. We could take the log of both sides and base to base 10 Natural log. I'm just going to use natural log, natural log of the left side and the natural log of this right side. And I'm not even gonna write down what that decimal is equivalent to. We would just put that in our calculator. I'll do that all at the end, and then we know log of a power will bump this guy down so we'll move that down. That becomes t over 25 and then we know T will equal this natural log. And I'm just gonna call that stuff A So we're gonna have the natural log of a whatever that is, we'll figure that out the minute and then I'd have to multiply by 25 and then I'd have to divide both sides by the natural law to. And let's figure out what that IHS So I have 25 times the natural log off, and I have my two million's of two and six zeros. 123456 divided by 322,000 066 And I'm gonna close my prophecy and then divided by natural log of to at your log of two. And again, I could have used whatever base I want. And when I do that, I get my time is pretty close to. I have 65.86 years. So about 66 years. So about 66 years from now from 2006. So if I wanted to find that year 2006, depending on the way the questions ordered, then I would have in about 2072 that we would have about two million bats, so probably wouldn't have many mosquitoes, since that's what they like to eat and we're finished

In 2000 for pain, initial population was 101 112,000. So consider this as an nor on the blink Time is 18 years. So consider this as it is, are doubling time. No. In party, we need to find a population model that is NT equals and north in the tourist A party by a there T is the time taken in years and is doubling time and nor these initial population Since we already know the value of and northern a therefore feeble substitute our value of and north it is 100. Well, well 112,000 and who is to party by is 18. So this is that equation one now for part B. Veneto form our equation model in exponential form. Therefore, for sleep even really to find our growth rate, which is our So you'll find, uh, we will first check what is the population Will 18 years has passed, So put big was 18 then our population model from equation one this implies and 18 equals 112,000 into tourist apart, 18 by 18. So this is 112,000 and go to since it inviting is one. Therefore, our population after two years, that is after 18 years is 200 24,000. Now, therefore, we know our population after 18 years is 224 1000. Therefore, we will put this value in exponential form off our model. So exponential form is and vehicles and north eaters to part R p and since then is 18. Our population is this. Therefore we will put the value off and he is eating on an 18 equals. And no, it is 112,000 in tow. It is two part are he's eating. So after putting value of an 18 we get this equation. No for being for our we find that this equation becomes two since if they divide this a question throat by 112 1 112,000 we get two week was it is two part it being are and taking long. Both sides forget Ellen do equals 18 are no solving for our We get I lento over 18. Now, using calculators, we can find the value off our which is 0.0 39 So this is our value of our that related growth read. Therefore, our population model is and the equals and north, which is 112,000 into eat, is two part odd which is 0.39 in 30. So this is our population mortal name this as a question number two nine Farsi we need to photograph. So firstly, they buy access as population in 2000 14 starting from 2014 is this is a population on This is our time in years. Therefore initially our population wars 112,000. So this is a first point. After 18 years, we found that our population becomes 224,000. Therefore, this is a population. After 18 years, that is 224,000. Now connecting these points, we get on population graph as this now in party we need to find that van Dillen population reached 1,500,000 so put the value off NTS 1500. So the 500,000 equals 112 We will put the value 500,000 integration number two and sold for P. Therefore dividing throughout by 112,000. We good? No e it is a part 0.39 p equals 4.46 taking low. Both sides figured 0.3 90 equals Alan 4.46 and now finally, for value off period will divide this equation throughout by 0.39 So after solving this, we get our value of P, which is 38.334 So therefore, a population Hillary's 500,000 after approximately 38 years. Therefore, it will take 38 years to reach 500,000.


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