Explanation 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: Limited growth model: d y d t = k y , k > 0 , t > 0 Calculation: Let y represent the annual sales. It is given that the rate at which the annual sales decline is proportional to the difference between the annual sales and $5 million. That is, d y d t = k ( 5 − y ) , k > 0 , t > 0 . Thus, the problem follows a limited growth model. Rewrite the differential equation d y d t = k ( 5 − y ) as, d y d t = k ( 5 − y ) d y 5 − y = k d t 1 5 − y d y = k d t Comparing the equation 1 5 − y d y = k d t and f ( y ) y ′ = g ( x ) , it is observed that f ( y ) = 1 5 − y and g ( t ) = k . By the above theorem, the solution is given by, ∫ f ( y ) d y = ∫ g ( t ) d t ∫ 1 5 − y d y = ∫ k d t ∫ ( 1 5 − y ) d y = ∫ k d t − ln ( 5 − y ) = k t + C 1 Simplify further as, e ln ( 5 − y ) − 1 = e k t + C 1 1 5 − y = e k t + C 1 5 − y = e − k t − C 1 5 − y = C e − k t , where C = e − C 1 That is, y = 5 − C e − k t . Initially the annual sale of a company is $0 million. That is, y ( 0 ) = 0 . Substitute t = 0 and y = 0 in y = 5 − C e − k t , y = 5 − C e − k t 0 = 5 − C e − k ( 0 ) C = 5 Substitute C = 5 in y = 5 − C e − k t , y = 5 − C e − k t = 5 − 5 e − k t The annual sale of the company after four years is $1 million

14th Edition

ISBN: 9780135961438

Author: Barnett

Publisher: PEARSON CO

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Chapter 9.2, Problem 67E

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To find: The sales and the rate of change of the sales during the fifteenth year, the time taken for the sales to grow to $4 million.

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Chapter 9.2, Problem 66E

Chapter 9.2, Problem 68E

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The sales and the rate of change of the sales during the fifteenth year, the time taken for the sales to grow to $4 million.

Solution Summary: The author calculates the time taken for the sales to grow to 4 million. The rate of change of annual sales increases by 0.12 million per year.

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To find: The sales and the rate of change of the sales during the fifteenth year, the time taken for the sales to grow to $4 million.

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