(a) Explanation Given: The culture of bacteria can support 1,100 bacteria. The rate of change is proportional to the product of the number present and the difference between 1,100 and the number present. The initial number of bacteria was 100. After 1 hour, the number of bacteria is 140. Theorem used: Separation of Variables: The solution of the differential equation f ( y ) y ′ = g ( x ) is given implicitly by the equation ∫ f ( y ) d y = ∫ g ( x ) d x . Formula used: Logistic growth model: If the rate of change with respect to time t of a quantity y is proportional to both the amount present and the difference between y and a limiting value M , then y satisfies the differential equation is, d y d t = k y ( M − y ) , k > 0 , t > 0 . Calculation: Let y represents the number of bacteria in the culture. The total number of bacteria in the culture is 1,100. The rate is proportional to the product of the number present and the difference between 1,100 and the number present. That is, d y d t = k y ( 1 , 100 − y ) , k > 0 , t > 0 . Thus, the problem follows a logistic growth model. Rewrite the differential equation d y d t = k y ( 1 , 100 − y ) as follows. d y y ( 1 , 100 − y ) = k d t 1 y ( 1 , 100 − y ) d y = k d t The given differential equation d y d t = k y ( 1 , 100 − y ) is in the form of f ( y ) y ′ = g ( x ) where, f ( y ) = 1 y ( 1 , 100 − y ) and g ( t ) = k . By the above mentioned theorem, the solution is given by, ∫ f ( y ) d y = ∫ g ( t ) d t ∫ ( 1 y ( 1 , 100 − y ) ) d y = ∫ k d t (1) Rewrite the integral ∫ ( 1 y ( 1 , 100 − y ) ) d y as, ∫ ( 1 y ( 1 , 100 − y ) ) d y = ∫ ( 1100 + y − y 1100 y ( 1100 − y ) ) d y = 1 1100 ∫ ( 1 y + 1 1100 − y ) d y Obtain the value of each integral as follows. 1 1100 ∫ ( 1 y + 1 1100 − y ) d y = 1 1100 [ ln y − ln ( 1100 − y ) ] Substitute ∫ ( 1 y ( 1 , 100 − y ) ) d y = 1 1100 [ ln y − ln ( 1100 − y ) ] in equation (1) and simplify as follows (B) To determine To find: The time taken for the culture to contain 1000 bacteria.

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 9.2, Problem 76E

(a)

To determine

To find: The number of bacteria present after 5 hours.

(B)

To determine

To find: The time taken for the culture to contain 1000 bacteria.

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Chapter 9.2, Problem 75E

Chapter 9.2, Problem 77E

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The number of bacteria present after 5 hours.

Solution Summary: Theorem used: Logistic growth model: if the rate of change with respect to time t of a quantity y is proportional to the amount present and the difference between 1,100 and