This post follows up on the question on Exam #3, regarding an exponential model for the world’s population. The first part is a review of the math we’ve discussed for exponential population growth (or decay) models. The second part discusses

Part I: Exponential Population Growth

This “mathematical model” of population growth assumes that the population in question grows (or decreases) at a constant rate over every time period. For example, on the exam, you were asked to assume that the world’s population is growing at a rate of 1.2% every year.

Given such a constant growth rate, we discussed why the resulting model is an exponential growth model, with the population at time t given by the function

where r is the constant growth rate (as a decimal, e.g., r = 1.2% = 0.012 on the exam question), and  is the initial population.

Hence, the function for the exam question, where the initial (current) population was given as 7 billion, is

(In the notation of the textbook,

, where c is the initial population.)

A typical exercise is to solve for the amount of time it takes for the population to double, in which case we need to solve the following equation for t:

We can solve for t do this by canceling the factor of, and then taking the logarithm of both sides and applying the log properties to “bring down” the variable t, so that we have

(Compare with the solutions to Exam 3, with the specific numbers for that exercise plugged in.)

Part II: World Population Growth Rates

(To be updated!  I’ll expand on the discussion we had in class on Monday, based on the data found on http://www.worldometers.info/world-population)