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Presented below in the table are the Australian population in mid-year 2006 and in mid-year 2016 by age rangesAustralian population (in thousand) 2006 3,937 13,273 ...

Question

Presented below in the table are the Australian population in mid-year 2006 and in mid-year 2016 by age rangesAustralian population (in thousand) 2006 3,937 13,273 2,644 19,854Australian population (in thousand) 2016 4,365 15.361 3,677 23,403Age range 0-14 15-64 65 and above TotalUsing the information provided (please keep at least 2 decimal places for all rates and ratios)_ project the size of the Australian population at the end of 2030, assuming the Australian population will grow at the same

Presented below in the table are the Australian population in mid-year 2006 and in mid-year 2016 by age ranges Australian population (in thousand) 2006 3,937 13,273 2,644 19,854 Australian population (in thousand) 2016 4,365 15.361 3,677 23,403 Age range 0-14 15-64 65 and above Total Using the information provided (please keep at least 2 decimal places for all rates and ratios)_ project the size of the Australian population at the end of 2030, assuming the Australian population will grow at the same rate as between 2006 and 2016 [5 marks] ii) project the size of Australian population aged 65 and above at the end of 2030, assuming the Australian population aged 65 and above will grow at 105% of its growth rate between 2006-2016. [5 marks]



Answers

In $1923,$ koalas were introduced on Kangaroo Island off the coast of Australia. In $1996,$ the population was $5000 .$ By $2005,$ the population had grown to $27,000,$ prompting a debate on how to control their growth and avoid koalas dying of starvation." Assuming exponential growth, find the (continuous) rate of growth of the koala population between 1996 and $2005 .$ Find a formula for the population as a function of the number of years since $1996,$ and estimate the population in the year $2020 .$

Living it to solve a problem about 27. Because a ears to be it here equals Yeah. So, first parties, Australia 20.9 equals 19.2 into here's to being do. Yeah, you know, the is he go to 0.0 8 40 which is 0.848 percent through a playoff deep because 19.2. And do you this too? 000848 Then why off 30 will be 24.765 Similarly can. And all Canada 34.3 equals that they weren't pointed three into us to be into. Yeah, because because several 0.915 So lay off deep. It was 31.3 in the years to 0.0 Man went five keep Then where you're 30 comes to be 41.19 then Philip. Thanks, Philip Points 95.9. Because 79.7 Hindu here is to being to 10. We will get because little 10185 So by off equals 79.7 in the years to 3.185 Then buy off that because 1 38.8451 38 point 84 keep South Africa 43.3 equals 44.1 in do us to be in the tent they got be with minus 0.183 which is why off peak equals 44.1 in the U. S. To when a 0.18 three What else 30 equals 41.743 then to keep 73.3. It was 65.7 in the years to be in the thank we got because 0.109 to playoff d equals 65.7 in the U. S. To 0.109 way off. 30 because 91.24 be purposed Australia and Turkey are growing at the rate off pointed for it person and 1.9 opportunity. This is Australia and this keep Bura presenter group her presence the group hi at the B I at the growth trick and like the value of B means shrinking off population click South Africa vehicles minus 0.183 Canada Because 0.0 91 ft growth trade or shrinking rate is given by big like do very little South Africa reflect city like to growth in population and puts two growth off Canada represents the was to growth for Canada. Thank you.

So how we're going to start with this is the population of Indonesia is P. That means that growth rate is dp DT. Um and that's going to equal K P work. It is a constant. So what we'll do is we will, uh, integrate both sides. So we integrate this side and then we'll integrate. This time, this is gonna be going from P one. You too. This will go from T one to t two. Okay, then when we integrate that, we'll end up getting is, uh, this is the one overpay BP, and this is we can pull out the k here and we'll just get DT when we integrate both sides. What will end up with is the natural log of key to over key to over P one equal to K times t two minus t one. And then we can isolate K. Um, so we know that K equals this whole thing. Times one over t two minus t one Santa K value there. Then if we substitute in the values of P one p two t one t two and t three. What we end up getting r p two. We want to be 1980. Um are the 1980 p one we know is 100 t one minus t two. Um, this is going to be 1960 1980 minus 1960. And then this right here is all going to be equal to natural log of 100 over 83. It's too s. What we end up getting is that p of 19 80 is equal to 145.16 approximately. So that tells us that the population of Indonesian 1980 would be approximately 145.16 million. And then since the actual population was 1 50 that means that, um, we were about 4.85 million from the predictable prediction, Um, which actually isn't too bad, given the large number there. Then for six b, we, um, want thio evaluate the population in the year 2000. So in order to do that, what we'll have is the natural log of T 2000 over the population that we already have 1 50 million times, one over 2000 minus 1980 and that's going to be set equal to the natural log of 1 50 over 100 and that is one over times one over 1980 in 1960. Then once we solve this for P 2000, we get that P 2000 equals 2 to 5, which tells us that the population of Indonesia in 2000 will be approximately 225 million. Um, since the actual population was 214 million, we see that we're allowed 11 million more than what actually happened. So our prediction was a little bit higher. Then we want to do part C, which is going to be predicting the 2010 population. We'll go through the same process as before. Um, this will be 2010 mhm Toby to 14, 2010. That's 2000. And then that will be equal. Thio 100 over 83 times, one over 2000, minus 1980 and out of that we get the 2010 give 2010 is 255.6 million. Um, and since the actual population was 243 million, we see that it was less than our prediction by about 12.6 million then lastly, for part D. We're going to do this one last time for 2020. So we're predicting it, um, of 2020 Um, in the 2000 result, which will be true. 14. Still, we'll have 2020 over 2000, and then we'll keep the same numbers as before. What we end up getting is that 2020 we'll have a population in Indonesia will have a population of 305 2.31 million. Um, so what we see is that it will be the estimate will be lower than the actual on. Because the interval is large, the prediction will be much more different than the actual, so the prediction will most likely be too low.

And this problem were given a bunch of infant mortality data from 1980 to 2000 and five. So we're we're told to basic set in 1980 as year zero. So I said he was zero and then the next was 90 85 90 the 95 2000 and 2005. So every five years we have some data. And since we want a quadratic fit, what we want to both they ask us for your find that at least squares lined with data. Um and also at least quadratic. So both both both a line and a quadratic function they asked us for. So I made a column of X squared values. And again um much problem video for problem 24 to get details of why I did this. So I put these in and I got and I squared each one of them. Obviously you can see. And then here's the the Y. Data. So the infant mortality rate and then to get the quadratic fit, I am I just did I did this uh he's the uhh the new estimate function on X and X squared. So to get Y as functions of both linear combination of X and X squared. And we see that the grid this for A. This for B. And this for C. And this expression here. And then for the linear one. I just did a linear estimate on X. But Y and X. Where you see there's no X squared here because I didn't I didn't want X squared in my fit. So here we can see here those two things now. What we can do if we uh if we plot the data uh until we plot the data and this same date over here and here's a quadratic fit. A second order polynomial. And we can see here the coefficients are the same as what we got here. Although this is an extra significant digits that obviously I could have, you know, increased the number of significant digits here. Let's actually just do that and see what let's see here a number plus. So there we go. You can see that they basically exactly match up. Wow. Um So what we have, then we can see this quadratic fit goes pretty good because you can see that this data kind of tapers off. And in fact we would expect that there's going to be some infant mortality, no matter what we do. So no matter how good medical practices we have, there's going to be some. And so we can see that this drop, but it may have may have leveled out and there might be, it might being a whole lot of ways we can get it below this value here. The linear fit. It looks pretty linear, but you can really see this trend of it leveling out here. So I would say that this is not a accurate fit. This is not good. This is a much better approximation here. And what the other thing you have to worry about is eventually this instance is a parappa. This thing is going to start going up. So if you extrapolate from here too far, So say you go out to maybe 40 years, you're gonna see that it starts going up again, which means that basically your way outside of the range of where this this model is valid. So let's see here they ask us. Um let's see here in 19 to estimate for in 2010. So that would be year 30. So I plugged in access. So I said, this is access 30. Then down here, I basically used the coefficients, A, B and C. So I have a X. Squared plus B, X plus E. And that tells us that we would expect about about 6.83 Hey? Um And again, it's kind of going up a little bit and that's probably, you know, again, not a good to extrapolate here because it's probably just plateau and it's probably going to kind of let you know, breach a limit of maybe 65 But what we can see the linear fit over here. Um If we go out here to 30 we see that we're down, we're way down at 4.78 as our infant mortality rate. Which seems like it's basically unless this was a real outlier of a year, um that seems they could probably not true. It's probably not going to get down to here because again, it looks like it's leveling off.

F. F. P. Equal to 85 upon one plus 1.859 time into the power negative 0.66 time T. Where t better than an equal to zero T. Less than equal to five. So here in the execution the U. S. Population aged 65 years and older from 2000 to 2050 is projected to grow at the red off half of equal to 85 upon one plus 1.859 time. Eat the power negative 0.66 time T. Where t better than an equal to zero T. Less than equal to five were T. Measured indicates with equal to zero corresponding to 2000. Bye. How much we will the population aged 65 years and older increase from the beginning of 2000 until the beginning of 2030. So here definite intrigue. A lot of F. F. T. Didi from 0 to 3 equal to and you drive it off 85 upon one plus 1.8 59 time. It the power negative zero point 66 times D. Duty from 0 to 3. So here multiply the integrated by it to the power 0.66 time T. So here we get definite intrigue. A lot of 85 time into the pole. 0.66 time T. Upon into the power 0.66 time T. Plus 1.859 DT from C. 023 So here we have to find the anti davidoff F. So here we use substitution method. So we take U. Equal to 1.859 plus each of the power 0.66 time T. So here we have to differentiate both sides with respect to T. So here we get Do you upon the X. Sorry didi equal to 0.66 time into the pole 0.66 time T. So now you can write like that do you upon C. 0.66 equal to into the pole 0.66 times T. Duty. So now we put here that value. So we're here we get in triangle 85 upon 0.66 times. Do you upon you? So here we get 85 upon 0.66 time. Natural. Look you plus C. So here you can see that the value of you is 1.859 plus eight. The ball 0.66 time T. So we put here and here we get 85 upon 0.66 time Natural look more Dallas 1.859 plus A. To the power. 0.66 times T. Plus. See now we put that value here. So we get definite integral of 85 time into the power zoo point 66 times T upon 1.859 plus into the power 0.66 time T DT from 0 to 3. So here we get the anti Israel F is 85 upon ceo 0.66 time Natural Morvillo's 1.859 plus. Into the power 0.66 times T from 0 to 3. So here offer value is three. So we put here equal to three. So here we get 85 upon 0.66 Natural models 1.859 plus. Into the power 0.66 times three. Sorry, upon 0.66 times Natural model. S Now here we have to put equal to zero. So marcellus 1.859 plus. Eat the power zero. So now we simplify this. And here we get 85 I 1.15 79 upon 0.66 equal 100 49 point 14 million. So here we can see the population aged 65 or more will be increased by 100 49.1 36 millions from 2000 to 2030. So it is a final answer.


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