(A) Explanation Given: The percentage of users aware of the brand is 10%. After the first week, 20% of the consumers are aware of the brand. The rate at which the people learn about the brand is proportional to the product of the percentage of informed consumers and the percentage of uninformed consumers. 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 d y d t = k y ( M − y ) , k > 0 , t > 0 .” Calculation: Let y represents the percentage of the informed consumers. The maximum percent is 100%, which is equal to 1. Thus, the number of people who are unaware of the brand will be 1 − y . The rate is proportional to the number of product of the percentage of informed consumers and the percentage of uninformed consumers. That is, d y d t = k y ( 1 − y ) , k > 0 , t > 0 . Thus, the problem follows a logistic growth model. Rewrite the differential equation d y d t = k y ( 1 − y ) in the form of equation f ( y ) y ′ = g ( x ) as follows. d y d t = k y ( 1 − y ) d y y ( 1 − y ) = k d t 1 y ( 1 − y ) d y = k d t Compare the equation 1 y ( 1 − y ) d y = k d t with equation f ( y ) y ′ = g ( x ) . It is observed that, f ( y ) = 1 y ( 1 − y ) g ( t ) = k By the theorem, the solution is given by, ∫ f ( y ) d y = ∫ g ( t ) d t ∫ 1 y ( 1 − y ) d y = ∫ k d t ∫ ( 1 y + 1 1 − y ) d y = ∫ k d t ln y − ln ( 1 − y ) = k t + C 1 Simplify further as, ln ( y 1 − y ) = k t + C 1 e ln ( y 1 − y ) = e k t + C 1 ( y 1 − y ) = e k t + C 1 y = ( 1 − y ) e k t + C 1 Simplify further as, y = C e k t − C e k t y , where C = e C 1 y + C e k t y = C e k t y ( 1 + C e k t ) = C e k t y = C e k t 1 + C e k t That is, y = C e − k t + C (B) To determine To find: The percentage of consumers aware of the brand name after 5 weeks, if 25% of the consumers are aware of the brand name after first week.

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

15th Edition

ISBN: 9781323047620

Author: Barnett, Ziegler, Byleen

Publisher: Pearson Education

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Chapter 9.2, Problem 62E

(A)

To determine

The percentage of consumers will be aware of the brand name after 5 weeks of advertising.

(B)

To determine

To find: The percentage of consumers aware of the brand name after 5 weeks, if 25% of the consumers are aware of the brand name after first week.

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Chapter 9.2, Problem 61E

Chapter 9.2, Problem 63E

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Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

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The percentage of consumers will be aware of the brand name after 5 weeks of advertising.

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 number of product of the percentage of informed

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The percentage of consumers will be aware of the brand name after 5 weeks of advertising.

To find: The percentage of consumers aware of the brand name after 5 weeks, if 25% of the consumers are aware of the brand name after first week.

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