Since the new census numbers are out, we want to highlight the modeling of the U. S. census. This question examines three models for studying the population of the U. S. during the 20^th century. Below is a table of the U. S. census data for the 20^th century.

a. The average growth rate for the 20^th century is 14.15%. Use the discrete Malthusian growth model

with P₀ = 75,994,575 and r = 0.1415 to simulate the population from 1900 to 2020. Make a table showing the actual population, the simulated values, and the percent error between the model and the actual data for the years 1930, 1950, 2000. What is the maximum percent error (in absolute value) for this model (over the range simulated) and when does it occur? Compute the average percent error (in absolute value) between the actual data and the model for the dates from 1910 to 2000.

b. Throughout U. S. history, immigration has played an important role. During the 20^th century, it has been tightly regulated and maintained a relatively constant value. Suppose that the immigration rate is m = 3,200,000 people per decade. The discrete Malthusian growth model with immigration is given by

where P₀ = 75,994,575 and r = 0.1150. Simulate this model from 1900 to 2020, and make a table showing the actual population, the simulated values, and the percent error between the model and the actual data for the years 1930, 1950, 2000. What is the maximum percent error (in absolute value) for this model (over the range simulated) and when does it occur? Compute the average percent error (in absolute value) between the actual data and the model for the dates from 1910 to 2000.

c. The two previous models grow without bound. One question is where the U. S. population will level off, and there are many estimates on what this might be. We studied the logistic growth model and found that it has this property of leveling off at the carrying capacity of the population. A study indicates that a good logistic growth model for the population of the U. S. in the 20^th century is given by

Again let P₀ = 75,994,575 and simulate this model from 1900 to 2020. Make a table showing the actual population, the simulated values, and the percent error between the model and the actual data for the years 1930, 1950, 2000. What is the maximum percent error (in absolute value) for this model (over the range simulated) and when does it occur? Compute the average percent error (in absolute value) between the actual data and the model for the dates from 1910 to 2000. Compute the carrying capacity for this model. Also, determine how long, according to this model, it will be until the population reaches 90% of the carrying capacity.

d. Graph all three models and the census data on the interval 1900 to 2020. Looking at the three models above, determine which model you believe best predicts the population for the years 2010 and 2020. Which model do you believe is the best and why? Describe two ways that you could improve the best model to make a better prediction for either the 2010 census or determining the carrying capacity for the U. S.