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The table gives estimates of the world population, inmillions, from 1750 to $2000 :$$\begin{array}{|c|c|c|c|}\hline \text { Year } & {\text { Population }} & {\text { Year }} & {\text { Population }} \\ \hline 1750 & {790} & {1900} & {1650} \\ \hline 1800 & {980} & {1950} & {2560} \\ \hline 1850 & {1260} & {2000} & {6080} \\ \hline\end{array}$(a) Use the exponential model and the population figuresfor 1750 and 1800 to predict the world population in1900 and $1950 .$ Compare with the actual figures.(b) Use the exponential model and the population figuresfor 1850 and 1900 to predict the world population in$1950 .$ Compare with the actual population.(c) Use the exponential model and the population figuresfor 1900 and 1950 to predict the world population in$2000 .$ Compare with the actual population and try toexplain the discrepancy.

a)$$ 1508 ~million$$$$ 1871~ million$$b) $$2161 ~million$$

c)$$ 3972 ~million$$

Calculus 1 / AB

Chapter 3

INVERSE FUNCTIONS

Section 4

Exponential Growth and Decay

Derivatives

Differentiation

Applications of the Derivative

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Hey, guys, welcome back. So this first problem were asked to observe some data in 17 50 1800 and then make some predictions for the year in 1919 50 so we can arbitrarily set t is equal to zero in the year. 17 50. Then we know for this condition right here he AT T is equal to zero is equal to 790. That would be in millions. And then we know that in the year 1800 population is 980 million sweetened right that p for the year 1800. Or that's when t would be 50 980. Because it is an exponential growth problem, we'll follow the form DP over. DT is equal to R constant K times P or population. Then if you integrate both sides, that becomes our population is equal to our initial population. Times e raised to the Katie so we can plug into values here and solve for K. So at T 0 to 50 we know that P is equal to 980. We have 980 zeal to our initial population, which is 790 times e raised. The Katie Que is unknown and rich he is 50 in this case is that released the 9 80 We can divide both sides by 7 90 and then take the central logger both sides and we're left with natural law of 9 80 over 7 90 the natural log of e raise to the 50 k not long, and you'll cancel out in her left with natural log. 9 80 over 7 90 See below 50 k Divide both sides by 50 and we get that K is equal to the natural log. 9 80 over 7 90 all over. And that comes out 2.431 Now we have the equation. P C go to 7 90 our initial population. Times E raised the 0.0 431 an arrest to find the population in the year 1900. Well, in the year 1900 that is one little T will be equal to 150 as 150 years after 17 50 we can plug in 1 50 into our equation. So we get P is able to 7 90 times e to the 0.431 times 1 50 And that comes out to 15 0 wait. Our population is 15. 08 million. Our estimation for the year 19 Now, rest estimated in 1950. So, in 1950 that is when t will be approximately will be 200. That's 200 years after the year. 17. 50 Now into the same thing and just plug in 200. So P is able to 7 90 the race to the 0.0 431 times 200. And when you that problem, you get a answer very close to 1871 So our answer is 18 71 1,000,000 for the year 1950. I believe that concludes Part A, which we want to part B now and in part B. We want to use the data in 18. 50 in the 1900 predict the population in 1950. So we're not gonna said T is equal to zero. T is equal to zero in the year 18. So we know that the population in 18 50 is 12 60 Antonio the population in 1900 just 50 years after that soapy of 50 Ziegel to 16 50. These are both in millions again. We'll go very similar process to the last party. So we'll say that 16 50 dessert p zeal to our initial value of 12. 60 times e raised to the K I'm 50. So again we could divide both sides by 12 60 then take the natural log of both sides and her left with natural law 16. 50 over 12 60 just like above the natural log and that you will cancel out and were left with 50 k We could say that K is equal to the natural log of 16. 50 over 12. 60 all over 50. And we do that on a calculated that comes out to 0.0 539 We can stop there. So now we have the equation. P is equal to 12. 60 Well, 60 times erase the 0.0 539 He and restaurant the population in the year 1950. So for 1950 that is 100 years after 18 50. So t is only 100 and we want to plug in 100 into this equation way of P. is equal to 12 60 times e race, the 0.0 539 times 100. And upon doing that calculation, you should get a value of very close to 2000 161 something right around there, 2161 1,000,000 for 19. We can see that this estimation is a lot higher than our estimation. In part I of 18. 71 million, we can see the actual data just comparing the actual data. For part, be actual data for the year 1950 is 25. 60 1,000,000 in 1950. See that both these estimations are actually lower than the actual data in 1950. Finally, in part C, we want to use the data in 1919 50 to figure out what the population will be in the year 2000. So in 1900 that's what we can Set T is equal to zero in Tuz zero in the air. 1900 population of that year, according to our charter is 16 50 and then population in 1950 nearly 50 years after that is approximately 2000 560 in the very same things apart and be confined. K 12 p 25 60 25 60 is equal to 16. 50. Sorry, my Penn is acting up a bit 16. 50. He raised to the 50 k Then we can divide both sides by 16. 50. Anything natural log on both sides were that the natural log 25. 60 over 16 50 is able to that not too long On the eternal cancel out Never left of 50 K we divide both sides by 50 and K is equal to the natural log 25 60 Over 16. 50 all over 50 The calculation of faster We get a answer of 0.0 878 Now we want to figure out what the population will be are predicted for the year 2000 in the year 2000 and in the year 2000 That is 100 years after 1900. So we can say that t will be 100. A great one. Small step frente Regina equation. We know that p be equal to our initial value, which he said was 16 50 times e race to the Katie. Okay, we said was 0.0 It's 78 in our TVA, and now we can plug in. Our teas is equal to 100. This must be 16 50 times E raised to the 0.0 878 times 100. And when you that calculation you get a value right around 3000 900 and 72 million 3000 900 72 million for the year 2000 is that the actual population? Actual population 2000 according to our chart, is six 1000. 80 million is a lot higher. Then our 3972 estimated there has to be some discrepancy there, and one possible explanation for that is me. Because all the wars in the first half of the 20th century actually increased the life expected. This increased the life expectancy in the second half of the century, So four people lived in the second half of the 20th century than predicted, based on all the wars that occurred right around 1930 1940. That is why the actual population is a lot larger. That, with the estimated population, is from our graph. And that concludes parts A, B and C of this problem that wraps things up. We learned something and thanks for watching

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