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Use differential equations to analyze given census data

by RoRi

Population figures for the United States in given years are given by:

    \[ \begin{array}{c | c} \text{Year} & \text{Population (in millions)} \\ \hline 1790 & 3.9 \\ 1800 & 5.3 \\ 1810 & 7.2 \\ 1820 & 9.6 \\ 1830 & 12.9 \\ 1840 & 17 \\ 1850 & 23 \\ 1860 & 31 \\ 1870 & 39 \\ 1880 & 50 \\ 1890 & 63 \\ 1900 & 76 \\ 1910 & 92 \\ 1920 & 108 \\ 1930 & 122 \\ 1940 & 135 \\ 1950 & 150 \end{array} \]

  1. Use the formula from previous exercise,

        \[ M = x_2 \frac{x_3 (x_2 - x_1) - x_1(x_3 - x_2)}{x_2^2 - x_3 x_1}, \]

    to find a value for M based on the data from the years 1790, 1850, and 1910.

  2. Repeat part (a) using the data from the years 1910, 1930, 1950.
  3. From parts (a) and (b) would we accept or reject the hypothesis that the equation in this previous exercise (Section 8.7, Exercise #13) is a valid growth law for the population of the United States?
  1. Using the formula we derived in the earlier exercise (linked above) we compute,

        \[ M = (23) \cdot \frac{92(23-3.9) - 3.9(92-23)}{23^2 - (3.9)(92)} = 201 \text{ million}. \]

  2. Again, we use our earlier formula and compute

        \[ M = (122) \cdot \frac{150 (122-92) - 92 (150-122)}{122^2 - (92)150)} = 217 \text{ million}. \]

One comment

  1. tom says:

    Part c). didn’t get answered so I will hazard a guess- the model is not consistent because for the values M chosen the initial value M/2 is larger then the census data. So based on this the model would be consistent after 1920 or thereafter.

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