Question

Consider the following Year Population (in millions) 1790 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 3.929 5.308 7.240 9.638 12.866 17.069 23.192 31.433 38.558 50.156 62.948 75.996 91.972 105.711 122.775 131.669 150.697

Answer 1

Consider the following census data for the united states between 1790 and 1950 are given in the table above construction a logistic population model using the data from 1790 1860 and 1980 Assume units is years since 1791 and p is population in millions a) Accordingly to the given data the logistic model is given as dc/dt = p(a - bp) solution is p(t) = ap0/ bto + a+b40e^-at with p(0) = 3.9819 (year,1790) p (u_0) = 12.866 (year,1830) p (110) = 75.996 (year 1900) sub p0 = 3.929 in below equation to get implies p (t) = ap0/bpot (a-bt_0)e^at = 3.929a/ 3929b +(1-39296)b^-at = a/b/1 +(1/3.929 -a/b-1)e^-at \r\n let consider c = 1/b, implies p(t) = c/ 1 +(c/3929-1)e^-at p (u_0) = 12.866 implies 12.866 = c/ (1 +c/3.929-1)e^-409 c = 12.866 + 12.866 (c/3.929-1) e^-40a rightarrow (1) implies 12.866 e^-40a = 3.929 (c-12.866/c.3929 p (110) = 75.996 c = 75.996 (c/ 3.929 -1) e^-1109 rightarrow (2)