Lesson Plan for Activity #8
Exponential Models
Continuous Change Model


Lesson # 9

Overview:

Students fit a continuous change 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 continuous change model
  • Predict future populations based on equation
  • Compare predicted population to given estimates
  • Determine limitations of model
Notes to Teacher: 

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 continuous change 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:  A(1) = (8,305,000) e (0.017)(1) = 8,447,392

Population in 2001:  A(2) = (8,305,000) e (0.017)(2) = 8,592,225

Population in 2010:  A(11) = (8,305,000) e (0.017)(11) = 10,012,735

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 natural log (ln) 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) e (0.017)(t)

ln 1.44 = ln e (0.017)(t)

ln 1.44 = (0.017) t ln e        [note: ln e = 1]

ln 1.44 = (0.017) t

t = ln 1.44 / 0.017 = 21.4 years

So, the population will reach 12 million 21.4 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) e (0.017)(t)

ln 2 = ln e (0.017)(t)

ln 2 = (0.017) t ln e         [note: ln e = 1]

ln 2 = (0.017) t

t = ln 2 / 0.017 = 40.8 years

So, the population will double 40.8 years after 1999 or sometime in 2039, assuming the growth rate stays at 1.7% during that time (which it probably will not.)

An interesting extension activity can be found in the Additional Explorations section. Students can examine original (digitized) documents of Thomas Jefferson's census calculations. The calculations use logs and students can be challenged to explain the mathematics behind his calculations.