Explanation Given: The annual sales of a company were $8 million, initially. The annual sales after two years are $6 million. The annual sales decline at a rate proportional to the difference between the annual sales and $3 million. 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: If the rate of change with respect to time t of a quantity y is proportional to the difference between y and a limiting value M , then y satisfies the differential equation, d y d t = k ( M − y ) , k > 0 , t > 0 Calculation: Let y represents the annual sales. The rate at which the annual sales decline is proportional to the difference between the annual sales and $3 million. That is, d y d t = k ( 3 − y ) , k > 0 , t > 0 . Thus, the problem follows a limited growth model. Rewrite the differential equation d y d t = k ( 3 − y ) in the form of equation (1) as follows. d y d t = k ( 3 − y ) d y 3 − y = k d t 1 3 − y d y = k d t Compare the equation 1 3 − y d y = k d t with f ( y ) y ′ = g ( x ) . It is observed that, f ( y ) = 1 3 − y , g ( t ) = k . By the above theorem, the solution is given by, ∫ f ( y ) d y = ∫ g ( t ) d t ∫ 1 3 − y d y = ∫ k d t ∫ ( 1 3 − y ) d y = ∫ k d t − ln ( 3 − y ) = k t + C 1 Simplify further as, e ln ( 3 − y ) − 1 = e k t + C 1 1 3 − y = e k t + C 1 3 − y = e − k t − C 1 3 − y = C e − k t , where C = e − C 1 That is, y = 3 − C e − k t Initially, the annual sales of a company were $8 million. That is, y ( 0 ) = 8 . Substitute t = 0 and y = 8 in y = 3 − C e − k t and obtain the value of C . y = 3 − C e − k t 8 = 3 − C e − k ( 0 ) C = 3 − 8 C = − 5 Substitute C = − 5 in y = 3 − C e − k t , y = 3 − C e − k t = 3 + 5 e − k t The annual sales after two years are $6 million

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ISBN: 9780134856667

Author: Stocker

Publisher: VST

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Chapter 9.2, Problem 68E

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To find: The sales and the rate of change of the sales 3 years from now and determine the time taken for the sales to decline to $3.5 million.

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Chapter 9.2, Problem 67E

Chapter 9.2, Problem 69E

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The sales and the rate of change of the sales 3 years from now and determine the time taken for the sales to decline to $3.5 million.

Solution Summary: The author calculates the time taken for the sales to decline to 3.5 million. The rate of change of annual sales decreases by 0.36 million per year.

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To find: The sales and the rate of change of the sales 3 years from now and determine the time taken for the sales to decline to $3.5 million.

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