(A) Explanation Given: The rate of change of bird population is proportional to the difference between population and a lower limit 200. The initial number of bird was 1,000 and declined to 500 in 5 years. Theorem used: Separation of Variables: The solution of the differential equation, f ( y ) y ′ = g ( x ) (1) is given implicitly by the equation, ∫ f ( y ) d y = ∫ g ( x ) d x (2) Formula used: Limited growth model: d y d t = k y ( M − y ) , k > 0 , t > 0 Calculation: Let y represents the number of bird on an island. The rate is proportional to the difference between population and the 200. That is, d y d t = k ( y − 200 ) , k > 0 , t > 0 Thus, the problem follows a limited growth model. Rewrite the differential equation d y d t = k ( y − 200 ) in the form of equation (1) as follows. d y y − 200 = k d t Compare the equation 1 ( y − 200 ) d y = k d t with equation (1) and observe that, f ( y ) = 1 ( y − 200 ) , g ( t ) = k . By the above theorem, the solution is given by, ∫ f ( y ) d y = ∫ g ( t ) d t ∫ ( 1 y − 200 ) d y = ∫ k d t ln ( y − 200 ) = k t + C 1 e ln ( y − 200 ) = e k t + C 1 Simplify further as, y − 200 = C e k t , where C = e C 1 y = 200 + C e k t Given that, the initial number of bird was 1,000 and declined to 500 in 5 years. That is, y ( 0 ) = 1 , 000 and y ( 5 ) = 500 (B) To determine To approximate: The lower limit M to the nearest integer.

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Chapter 9, Problem 47RE

(A)

To determine

The bird population on an island after 4 years from now.

(B)

To determine

To approximate: The lower limit M to the nearest integer.

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Chapter 9, Problem 46RE

Chapter 9, Problem 48RE

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Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

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The bird population on an island after 4 years from now.

Solution Summary: The author explains that the rate of change of bird population is proportional to the difference between population and a lower limit 200.

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The bird population on an island after 4 years from now.

To approximate: The lower limit M to the nearest integer.

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Chapter 9 Solutions