If it is assumed that the earth cannot support a population greater than 20 bill

Question

If it is assumed that the earth cannot support a population greater than 20 billion people, and that the
rate of population growth is proportional to the difference between how close the world population is
to this limiting value. According to the U.S., & World Population Clock, as of 10/9/2012, the world's
population is 7,044,579,690.
A)What is the mathematical expression describing the world population as
a function at time t?
B)Using the model developed from part (A), predict the earth's population on 10/9/2022

Explanation / Answer

Let me start by saying that you do not have enough information to solve this problem. In the following, we allow K to represent 20 billion. (A) The mathematical expression describing "rate proportional to difference between the population and the limiting population" is given by dP/dt = r (K - P) The solution is P(t) = K + (P0 - K)e^(-rt) where P(0) = P0. If we set the clock to be zero at 10/9/2012, then P0 = 7,044,579,690. (B) To be able to find the population at t=20, just plug in t=20. P(20) = K + (P0 - K) e^(-20 r) But this is the point where you need an additional piece of information to be able to find the value of r. Once you know r, just plug it in to this expression.

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