Chapter 4.2, Problem LC

LINKING

concepts...

For Individual or Group Explorations

Modeling the U.S. Population

To effectively plan for the future one must attempt to predict the future. Government agencies use data about the past to construct a model and predict the future. The following table gives the population of the United States in millions every 10 years since 1900 (Census Bureau, www.census.gov).

a) Draw a bar graph of the data in the accompanying table. Use a computer graphics program if one is available.

b) Enter the data into your calculator and use exponential regression to find an exponential model of the form y = a • b^x, where x = 0 corresponds to 1900.

c) Make a table (like the given table) that shows the predicted population according to the exponential model rather than the actual population.

d) Plot the points from part (c) on your bar graph and sketch an exponential curve through the points.

e) Use the given population data with linear regression to find a linear model of the form y = ax + b and graph the line on your bar graph.

f) Predict the population in the year 2020 using the exponential model and the linear model.

g) Judging from your exponential curve and the line on your bar graph, in which prediction do you have the most confidence?

h) Use only the fact that the population grew from 76 million at time t = 0 (the year 1900) to 309 million at time t = 110 (the year 2010) to find a formula of the form P(t) = P₀ • e^rt for the population at any time t. Do not use regression.

i) Use the formula from part (h) to predict the population in the year 2020. Compare this prediction to the prediction that you found using exponential regression.