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Numerade Educator

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Problem 9 Easy Difficulty

The population of the world was about 6.1 billion in 2000. Birth rates around that time ranged from 35 to 40 million per year and death rates ranged from 15 to 20 million per year. Let's assume that the carrying capacity for world population is 20 billion.
(a) Write the logistic differential equation for these data. (Because the initial population is small compared to the carrying capacity, you can take $ k $ to be an estimate of the initial relative growth rate.)
(b) Use the logistic model to estimate the world population in the year 2010 and compare with the actual population of 6.9 billion.
(c) Use the logistic model to predict the world population in the years 2100 and 2500.

Answer

a)$\frac{1}{305} P\left(1-\frac{P}{20}\right)$ with $P$ in billions.
b) $$\frac{20}{1+\frac{139}{61} e^{-t / 305}}, \text { so }$$
c) 13.87 billion.

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Video Transcript

heads Clara. So onion, right here. So we know M is equal to 20 and K is equal to 220 over 1000 over a 6.1, which is equal to one over 305. So we know the different Schauble equation. To be deep key over. DT is equal to P over 305 times one minus p over 20. We got we get this equation and we're gonna find for a We get the carrying capacity week to be 20 minus p of zero to be six point on over 6.1. So we got about 2.28 So we got P of tea is equal to 20 over one plus 2.2 a. He to the negative. 0.0 33 p. And we're looking at the population in 2010. So we plug in 10 to get about 6.24 billion, which is less than the real population for part C. We got that. We're using the equation we got from part, eh? So we're looking at the population and 4100 to get about 7.5458 and in 2500 we just plugging 500 when we get 13.91