Explanation Given: The Malthusian population p ( t ) with a negative growth rate- d p d t = − k p ( 2 ) p ( t 2 ) − p ( t 1 ) is the number of individuals in the population whose lifetime lies between t 1 and t 2 The average lifetime of the population is given by the formula ∫ 0 ∞ t | d p ( t ) d t | d t p ( 0 ) Initial condition- p ( 0 ) = p 0 Approach: Solve differential equation. d p d t = − k p Use the following logarithmic properties, wherever required- ln ( x n ) = n ln ( x ) ln a b = ln a − ln b ln ( a b ) = ln a + ln b ln e = 1 Average lifetime T A v e r a g e of the population is given by- T A v e r a g e = ∫ 0 ∞ t | d p d t | d t ∫ 0 ∞ p ( t ) d t ( 1 ) Calculation: Solve equation ( 2 ) d p d t = − k p d p p = − k d t Integrate both sides of the above equation. ∫ d y y = ∫ − k d t ln | p | = − k t + ln C ln | p C | = − k t p ( t ) = C e − k t ( 3 ) Substitute initial condition in equation ( 3 )

7th Edition

ISBN: 9780321977175

Author: Nagle

Publisher: PEARSON CO

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Chapter 5.5, Problem 8E

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The average lifetime of the population is $1k$ .

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