(a) Explanation Formula used: The solution of the logistic equation d p d t = P ( a − b P ) is given as, P ( t ) = a P 0 b P 0 + ( a − b P 0 ) e − a t . Calculation: The Census data for the United States between 1790 and 1950 are given in Table 3.2.1 Year Population (in millions) 1790 3.929 1800 5.308 1810 7.240 1820 9.638 1830 12.866 1840 17.060 1850 23.192 1860 31.433 1870 38.558 1880 50.156 1890 62.948 1900 75.996 1910 91.972 1920 105.711 1930 122.775 1940 131.669 1950 150.687 Table 3.2.1 The Logistic equation is given by, d p d t = P ( a − b P ) The solution of the logistic equation is given below. P ( t ) = a P 0 b P 0 + ( a − b P 0 ) e − a t … … ( 1 ) Here, P 0 = 3.929 millions . Substitute the value of P 0 in equation ( 1 ) . P ( t ) = a P 0 3.929 b + ( a − 3.929 b ) e − a t = a b 1 + ( 1 3.929 a b − 1 ) e − a t Assume a b = c . Substitute a b for c in the above equation. P ( t ) = c 1 + ( c 3.929 − 1 ) e − a t … … ( 2 ) From the table 3.2.1, P ( 60 ) = 23.192 millions , this implies that, Substitute 23.192 for P ( 60 ) and 60 for t in the above equation. 23.192 = c 1 + ( c 3.929 − 1 ) e − a t c = 23.192 + 23.192 ( c 3.929 − 1 ) e − 60 a e − 60 a = c − 23.192 23.192 ( c 3.929 − 1 ) … … ( 3 ) Also P ( 120 ) = 91 (b) To determine The error and percentage error between actual census population and predicted population for each year.

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ISBN: 9781284127003

Author: ZILL

Publisher: JONES+BARTLETT LEARNING,LLC (CC)

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Chapter 2.8, Problem 4E

(a)

To determine

A logistic population model using the data from 1790, 1850, and 1910.

(b)

To determine

The error and percentage error between actual census population and predicted population for each year.

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Chapter 2.8, Problem 3E

Chapter 2.8, Problem 5E

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A logistic population model using the data from 1790, 1850, and 1910.

Solution Summary: The author presents a logistic population model using the data from 1790, 1850, and 1910.

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A logistic population model using the data from 1790, 1850, and 1910.

The error and percentage error between actual census population and predicted population for each year.

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Chapter 2 Solutions