Teacher's Guide for Activity #7
Exponential Models
Constant Growth Rate Model


Lesson # 8

Overview:

Students fit a constant growth rate exponential model to U.S. population data and use the model to predict future population.

Objectives: 

  • Write an equation to determine U.S. population based on constant growth rate model
  • Predict future populations based on equation
  • Compare predicted population to given estimates
  • Determine limitations of model
Notes to Teacher: 

If there is time available, have students record the national population each day for a week using the U.S. Population Clock. Students should examine the data to see which type of function best represents the data.

Using the general equation and data provided in this activity, students will determine a specific equation to express the population as a function of t, the number of years after 1999. Students can then plug in any year after 1999 into the model to find the predicted population. Students should compare their results to the estimates given by the U.S. Census Bureau and explain why there might be differences.

An example of how to use the constant growth rate model is shown below using population data from the Dominican Republic. This data can be found using the International Database (IDB) Summary Demographic Data web page.

Estimated population in 1999:  8,305,000

Estimated growth rate for 1990-2000: 1.7% = 0.017

Population in 2000:  f(1) = (8,305,000)(1.017)1 = 8,446,185

Population in 2001:  f(2) = (8,305,000)(1.017)2 = 8,589,770

Population in 2010:  f(11) = (8,305,000)(1.017)11 = 9,997,010

An optional activity is provided in which students are asked to determine in which year the population reaches a certain amount. This can be solved by taking the log of both sides of the equation.  An example using data from the Dominican Republic is shown below. For the purposes of this example, the question posed is:

Question: The Dominican Republic's estimated population in 1999 was 8,305,000 with an annual growth rate of 1.7%. If this growth rate remains constant, when will the population reach 12 million? When will the population double in size from its present value?

a) When will population reach 12 million?

12,000,000 = (8,305,000)(1.017)t

1.44 = (1.017)t

log 1.44 = log (1.017)t

log 1.44 = t log 1.017

t = log 1.44 / log 1.017 = 21.6 years

So, the population will reach 12 million 21.6 years after 1999 or sometime in 2020, assuming the growth rate stays at 1.7% during that time (which it probably will not.)

b) When will the population double?

2 (8,305,000) = (8,305,000)(1.017)t

2 = (1.017)t

log 2 = log (1.017)t

log 2 = t log 1.017

t = log 2 / log 1.017 = 41.1 years

So, the population will double 41.1 years after 1999 or sometime in 2040, assuming the growth rate stays at 1.7% during that time (which it probably will not.)