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Problem 6 Hard Difficulty

The table gives the population of Indonesia, in million, for the second half of the 20th century.

(a) Assuming the population grows at a rate proportional to its size, use the census figures for 1950 and I960 to predict the population in 1980. Compare with the actual figure.
(b) Use the census figures for I960 and 1980 to predict the population in 2000. Compare with the actual population.
(c) Use the census figures for 1980 and 2000 to predict the population in 2010 and compare with the actual population of 243 million.
(d) Use the model in part (c) to predict the population in 2020. Do you think the prediction will be too high or too low? Why?

Answer

a) Population of Indonesia in 1980 will be approximately 145.16 millions.
b) Population of Indonesia in 2000 will be 225 millions.
c) Population of Indonesia in 2010 will be 255.6 millions.
d) 305.3 Million. The estimate should be over because the world population growth rate is shrinking.

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Video Transcript

in this problem, we have the table that gives us the population in Indonesia over various years and what we want to do, in part A is used to of those points to create a model for the population growth and then use that model to make a prediction. So what we're going to do is use these numbers in this equation where P not is going to be 83 million and pft is going to be 100 million on the time that has elapsed between the year 1950 the year 1960 is 10 years. So we have 100 million equals 83 million times e raised to the 10 K that's using this equation and substituting the numbers in. And now we can solve for K, the growth constant. And once we have that, we'll be able to write our model. So we divide both sides by 83 we get 100 over 83 equals E to the 10 K. Then we take the natural log of both sides. So we have the natural log of 183rd equals 10 k, and then we divide both sides by 10 and we get K equals and natural log of 183rd divided by 10. So that goes into our model. And we have p of tea equals 83. The initial population Times E raised to the power of K natural log 183rd over 10 times a time. And that's the model we're going to use to make the prediction where in this model t equals zero is the year 1950. Okay, so we want to make a prediction for the year 1980 and the year 1980 would be t equals 30. So we're finding the population at time 30. Substitute 30 into the equation. Put it in your calculator and you get approximately 145.16 million. Now, from the table, we see that the actual population in that year was 150 million. So this value is a little lower than the actual value. Okay, now we're gonna move on to Part B and we do something very similar. We take the ordered pairs 1960 with a population of 100 million and 1980 with a population of 150 million we find the model again. Using PFT equals P not each of the K T. And then we use the model to make a prediction. So our final amount is going to be 1 50 Our initial amount is 100 the time elapsed here is 20 years, so we have 1 50 equals 100 times E to the 20 k, and we'll solve that for K. We divide both sides by 1 50 we get 1.5 equals e to the 20 k. Then we take the natural log of both sides and then we divide by 20 and we have our value of K natural log of one point 5/20. We substitute that into our model. We have p of t equals 100 times e to the natural log of one point 5/20 times teeth. And we'll use that model to make a prediction. And in that model, T equals zero Is the year 1960. So for this part of the problem were interested in making a prediction for the year 2000 and the year 2000 would then be t equals 40. So we're finding P of 40 we substitute 40 into our equation, put it in the calculator and we get approximately 225 million. Now the actual value was 214 million. So in this case, the model gave us a value too high. All right, we're going to go through this scenario one more time for part C. And we have the year 1980 with a population of 150 million. And we have the year 10,000 our 2000 with a population of 214 million. And we're going to use our model and find the value of K. So our final population is 214. Our initial population is 1 50 the time elapsed is 20 years. So we're putting 20 and 40. We'll solve this for K. We'll divide both sides by 1 50 it reduces to one or 7/75. Then we take the natural log of both sides. So the natural log of one of 7/75 equals 20 K, and then we divide both sides by 20 and we have our value of K natural log of one of 7/75 over 20. So we put that into our model and we have p of t equals 1 50 times e to the natural log of one or 7/75 divided by 20 times T Okay, Now, for this model, the time zero would be the year 1980 and the prediction were interested in making is for the year 2010. So that's going to be 30 years after, uh 1980. So we're gonna let t equal 30. So finding p of 30 and that is approximately 255.6 million now The actual value waas 243 million. So once again, the model gave us a value that's too high. And finally for party, we're going to use the same model that we got from part C, and we're going to make a prediction for the year 2020 which happens to be the year that I'm recording this video in 2020. The time would be 40 years since the year 1980. So we'll use a 40 in our population model and we get approximately 305.3 million as the population of Indonesia. Now what do we think? Do we think that's going to be too high, too low or what? Well, we saw in part C. We got a value too high. We saw in part B. We got a value too high. So chances are this one will be too high. It seems like the population growth is slowing.