Use the method of Problem 36 to fit the logistic equation to the actual U.S. population data

Question:

Use the method of Problem 36 to fit the logistic equation to the actual U.S. population data (Fig. 2.1.4) for the years 1850, 1900, and 1950. Solve the resulting logistic equation and compare the predicted and actual populations for the years 1990 and 2000.

Problem 36

To solve the two equations in (10) for the values of k and M, begin by solving the first equation for the quantity x = e^-50kM and the second equation for x² = e^-100kM. Upon equating the two resulting expressions for x² in terms of M, you get an equation that is readily solved for M. With M now known, either of the original equations is readily solved for k. This technique can be used to “fit” the logistic equation to any three population values P₀, P₁, and P₂ corresponding to equally spaced times t₀ = 0, t₁, and t₂ = 2t₁.