(a) If t is time measured in years, show that k = 0.03 results in an annual increase (or annual percentage growth rate) of approximately 3% per year, using equation (3).
(b) Plot the world population data in Table 1.1.1 on the same graph as the Malthus model.
(c) How might the Malthus exponential population model be modified to better fit the actual population data in Table 1.1.1? Make an argument for why the exponential function is not unreasonable for these data and identify what must change to fit the given data. This is a qualitative question rather than a quantitative one, so you need not find exact numbers.

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Table 1.1.1 Comparison of Malthus model y(t) -0.9e0.03 and actual world population (in billions). Year t Malthus Actual YeartMalthos Actnal 1800 181010 1820 20 183030 1840 40 185050 186060 187070 1880 80 1890 90 13.40 190010018.09 0.9 24.42 32.98 44.52 0 0.90 1.21 1.64 2.21 2.99 4.03 5.45 7.35 9.93 1910 110 1920 120 1930 130 1940140 60.10 1950150 1960160 109.53 1970 170 147.87 1980180199.62 1990190 269.49 2000 200 363.81 1.8 0.9 2.3 2.7 3.0 3.5 4.2 5.1 6.0 1.0 81.13 1.2 1.5 1.7