Modelling Exponential Growth - Using Logarithms

Let's consider the following problem.

Statistics indicate that the world population since World War II has been growing at the rate of 1.9% per year. Further, United Nations records indicate that the world population in 1975 was (approximately) 4 billion. Assuming an exponential growth model, (a) what will the population of the world be in the year 2000? (b) When will the world population be 7 billion?

(a) Let A be the world population (in billions) at time t, in years since 1975. Then A0 is the world population in 1975; therefore A0= 4 .
The rate of growth is actually the k value in the formula for exponential growth given above; for this problem, k=0.019. So the exponential model for this problem is

A = 4e0.019t

The year 2000 corresponds to t=25, so to estimate the world population at this time we evaluate the function at t=25:

A=4 e 0.019(25)= 4 e0.475 = 6.43 (approx.)


The model predicts a world population of 6.43 billion in the year 2000.

(b) For the other question, we are given the population and we want to determine the time when the population is the given size. So we set A=7 in the function (formula) for the population, and solve for time t.

7=4e0.019t

Isolating the exponential part gives
1.75= e0.019t.

Now, to solve for t, we take natural logs (ln) of both sides:
ln 1.75 = ln(e0.019t)

and the cancellation property for logarithms says
ln 1.75 = 0.019t.

Then solving for t gives
t= (ln 1.75)/0.019 = 29.4, approximately.

According to the model, the world population should be 7 billion around the years 2004 or 2005. (1975 + 29 = 2004, 1975 + 30 = 2005.)


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