Activity # 7

Activity #8 Exponential Models Continuous Change Model

Activity # 9

There are other functions that can be used to model exponential growth. If a population grows (or decays) continuously over a period of time such that the rate of change of population is proportional to the total population, then a continuous change model for exponential growth can be used. This model is similar to the model used to determine how much a savings account grows when the interest is compounded continuously.

This exponential model can be used to predict population during a period when the growth of a population is continuous. The constant growth rate model used in Activity 7 does not assume continuous growth. 

1. From the U.S. Census Bureau's Historical National Population Estimates, 1900 to 1999, record the national population for 1999 and the average annual percent change (growth rate given in percent) for that year.
   
   

2. Assuming that the growth rate remains constant, give an equation to express the population as a function of t, the number of years after 1999. Remember:

   A(t) = Population t years after 1999

   P = Population in 1999

   e = 2.718....

   r = Annual growth rate in decimal form

   t = number of years after 1999

   Example: In July 1964 the population was 191,888,791 and the annual growth rate was +1.39 % or 0.0139. An equation to find the population t years after 1964 would be: A(t) = (191,888,791) e (0.0139)t

    

3. Using the equation you determined for 1999 data, predict the population in 2000, the current year, and 2010.
   
   

4. Compare your results to the estimated values given in the International Database (IDB) Summary Demographic Data for the U.S. as well as the U.S. Population Clock. How close were your results? Why might they be different?
   
   

5. Do you think this continuous change model is any better than the constant growth model for predicting population in the long term? Support your answer by showing what today's predicted population would be if the continuous change model was used for a year in the past, assuming that the growth rate remained at the same value as in that year. 
   
   

6. Optional: Using the continuous change model, predict when the U.S. population will reach 300 million, assuming the growth rate remains at its present rate. When will it reach 350 million? When will the population double in size from its present value? Why might your results differ from those given in the International Database (IDB) Summary Demographic Data ?