The table gives the population of India, in millions, for the
second half of the 20 th century.
$$\begin{array}{|c|c|}\hline \text { Year } & {\text { Population }} \\ \hline 1951 & {361} \\ \hline 1961 & {439} \\ \hline 1971 & {548} \\ \hline 1981 & {683} \\ \hline 1901 & {846} \\ \hline 2001 & {1029} \\ \hline\end{array}$$
(a) Use the exponential model and the census figures for
1951 and 1961 to predict the population in $2001 .$ Com-
pare with the actual figure.
(b) Use the exponential model and the census figures for
1961 and 1981 to predict the population in $2001 .$ Com-
pare with the actual population. Then use this model to
predict the population in the years 2010 and 2020 .
(c) Graph both of the exponential functions in parts (a)
and (b) together with a plot of the actual population.
Are these models reasonable ones?

Answer

a) $$960~million$$
$$
6.6 \%
$$
b)$$ 1063~million$$
$$3.3\%$$
c) $$1297~ million$$
$$1617~million$$

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Calculus 1 / AB

Essential Calculus Early Transcendentals

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INVERSE FUNCTIONS

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Exponential Growth and Decay

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Video Transcript

you guys welcome back. Have another population problem here per day. You want to look at the years 1951 and 1961 to figure out the population in 2001. Let's let t equals zero in the air 1951 and we can say that the population T equals zero. According Dorine data is 361 and this data will all be in millions. Then we all say that p and looking at the year 1961. Now that is 10 years after 1951. So agencp of 10 is equal to 439. Because it is an exponential growth problem, we know they will follow differential equation dp over DT zeal to KP or P to the initial population. Times e to the k t. It sounds plug ins and values and figure out what K is. You know that p of tennis 439 blood that in 439 it's legal to our initial population, which we found was 361 Im e raised to the Katie is unknown and T is 10. That 10 relates the 439 million divide both sides by 361. In the natural log, we have the natural log, the 4 39 over 3 61 equal to the natural log in the eternal cancel out or left with 10-K So that's K is just the natural log before 39 over 3 61 all over 10. And when you do that math, you get a value very close to 0.0 19 56 Now we know that we have the equation. He eagle to our initial population. 361. He raised the point 019 56 Where t is zero in the year 1951. I want to predict the population and 2001. So in 2000 and one year 2001 that is exactly 50 years after 1951. That'll be won t the 50 50. Your brother. He is equal to 50. It was plugging 50 in our equation. See, we get so we have P is equal to 361 times E to the 0.1956 multiplied by 50 And then when we do that, we get a value of 960. That will be 960 million for estimation from population in the year 2000 and one. We can see that our actual population, actual population in 2001 actual 2000 in one population with 1000 and 29 million according to our chart. So if we wanted to, we could find the percent difference should be actual minus the theoretical divided by the actual absolute value that I'm 200% going to that we have are actual value was 129 1,000,000. I theoretical was 960 million. The actual was 1000 and 29 with that all by 100%. Then we get a value of 6.6% 6.6%. That is how much percent air we have. Part B. You want to use the data in the years 1961 and I can anyone and figure out what it is in 2001. Now let's say that T is equal to zero in the air. 1961. So according to our chart then we can say population in 1961 or P of zero, 439. And then we can also say population in 1981 will be 20 years after that. So p 20 in 1981 that value can also take from the table. And that is 600 and 83 600 83. But he was a very similar process. We didn't part a really have the equation. P is able to initial population times each of Katie unplugging our values. 683. It's legal to 439. Okay, dedicate multiplied by 20. We can then divide both sides by 439. Then we'll take the natural log of both sides. So what do that We get the natural log 6 83 over 4 39 Evil to the After logging, the eternal cancel out enter is left with 20 K on the right hand side. Now we can solve for K. We know that K. Is he going to the after law of 6 83 for 39 all over 20. That comes out to a value of 0.0 2 to 1. Now that we have our equation, we can write our new equation filled out with her variables filled in P is equal to 439 times E 2.2 to 1. Now we want to figure out what our population is in the year 2000 and one well, in the year 2000 and one year, 2000 and one t will be 40. We can say that population in 2001 population AT T 0 40 is equal to 439 times e to the 0.0 2 to 1 multiplied by 40. And when we do that calculation, we get a value of 1000 in 63 that also be in millions. Of course, because he had that compares to our actual value in the actual value actual value was it above was 1000 and 29 1,000,000. So again, we're gonna find a percent air. So we can say will be our same actual mice theoretical over the actual time, 200% we'll have the absolute value of 1029 minus 1063 over 1029 multiplied by 100% and that comes out to 3.3%. So we can see that our air value is smaller in our error value in party. Next. We want to use this model to predict the population in the years 2010 in 2000 20. So if we love our same equation p 0 to 4 39 e to the 0.0 2 to 1 time see so for the year 2000 in 10 we can see that would be 49 years after 1961. So TV 49 Now you just plug in 49 into our equation. Upon doing that, we had P Eagle 2439 for 39 eat to the 0.2 to 1 multiplied by 40 5439 times E to the 0.2 to 1 time 40. And that gives us the value of 1000 and 63 million about population in 2010. And then we're also has to figure out what the population might be in the years 2000 and 20. So in the year 2020 3 10 years after 2010. So we can say that T will be 59. We'll have the same equation. P is equal to 439 He raised the 0.2 to 1 times 59. And then when you that equation, you get a value of 16. Sorry. 17 1,000,000. My apologies. I realized I had one small error in this guy right here. This should not be 40. There should be t 0 to 49 and that will change your answers lately. So our new answer actually should have been 12. 96 12. 96 million for the year 2010 and then 16. 17 million for the year 2020. Then, finally, in part, C arrested Graf both of our equations. I went ahead and I grafted in gizmos shown here and expand it. So this was our population equation From the beginning of the problem. Our day and this is our graphing equation of blue line in part B. And we could see that both of those equations look pretty reasonable population functions. So thanks for watching hope this help

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