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Use differential equations to determine a population growth constant and predict future population

by RoRi

Using the census data provided in the previous exercise (Section 8.7, Exercise #16):

  1. Plot the graph of \log x as a function of time, where x is the population. Use this to show that the simplest growth law:

        \[ \frac{dx}{dt} = kx, \]

    was nearly satisfied from 1790 to 1910 and determine a value of the constant k.

  2. Determine a value of the constant k from 1920 to 1950, and assume that this value will hold in the future. Use this to estimate the population of the United States in 2000 and 2050.
  1. Rather than a plot, here is the table of values of \log x at each value of t:

        \[ \begin{array}{c | c | c} \text{Year} & \text{Population (in millions)} & \log x \\ \hline 1790 & 3.9 & 1.4\\ 1800 & 5.3 & 1.7\\ 1810 & 7.2 & 2.0\\ 1820 & 9.6 & 2.3\\ 1830 & 12.9 & 2.6\\ 1840 & 17 & 2.8\\ 1850 & 23 & 3.1\\ 1860 & 31 & 3.4\\ 1870 & 39 & 3.7\\ 1880 & 50 & 3.9\\ 1890 & 63 & 4.1\\ 1900 & 76 & 4.3\\ 1910 & 92 & 4.5\\ 1920 & 108 & 4.7\\ 1930 & 122 & 4.8\\ 1940 & 135 & 4.9\\ 1950 & 150 & 5.0 \end{array} \]

    This indicates that \log x is approximately linear over this time period. We can compute

        \[ k_{\text{avg}} = \frac{\log 150 - \log 3.9}{1950 - 1790} = 0.026. \]

  2. The average value of k from 1920 to 1950 is

        \[ k_{\text{avg}} = \frac{\log 150 - \log 108}{1950 - 1920} = 0.011. \]

    This implies

        \begin{align*} x(2000) &= \exp (\log 150 + 0.011 (2000 - 1950)) = 257 \text{ million}. \\ x(2050) &= \exp (\log 150 + 0.011 (2050 - 1950)) = 446 \text{ million}. \end{align*}

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