(A) 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: “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 S represents the annual sales of the company with the initial condition S ( 0 ) = 0 . In first year, the sale of the company is $50,000. The rate of growth of annual sales of a company is proportional to the difference between annual sales S and $200,000. This implies, d S d t = k ( 200 , 000 − S ) , k > 0 , t > 0 Thus, the problem follows a limited growth model. Rewrite the differential equation d S d t = k ( 200 , 000 − S ) in the form of equation f ( y ) y ′ = g ( x ) as follows. d S ( 200 , 000 − S ) = k d t 1 ( 200 , 000 − S ) d S = k d t Compare the equation 1 ( 200 , 000 − S ) d S = k d t with equation f ( y ) y ′ = g ( x ) . It is observed that, f ( S ) = 1 ( 200 , 000 − S ) g ( t ) = k By the theorem, the solution is given by, ∫ f ( S ) d S = ∫ g ( t ) d t ∫ ( 1 200 , 000 − S ) d S = ∫ k d t − ln ( 200 , 000 − S ) = k t + C 1 ln ( 200 , 000 − S ) − 1 = k t + C 1 Simplify further as, e ln ( 200 , 000 − S ) − 1 = e k t + C 1 ( 200 , 000 − S ) − 1 = e k t + C 1 200 , 000 − S = e − k t − C 1 S = 200 , 000 − C e − k t , where C = e − C 1 Given that, S ( 0 ) = 0 . Substitute t = 0 and S = 0 in S = 200 , 000 − C e − k t and obtain the value of C (B) To determine To find: The sales of first year if the sales must to be achieve $150,000 after 3 years.

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, Problem 41RE

(A)

To determine

To find: The time required to reach the annual sale of $150,000.

(B)

To determine

To find: The sales of first year if the sales must to be achieve $150,000 after 3 years.

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Chapter 9, Problem 40RE

Chapter 9, Problem 42RE

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

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The time required to reach the annual sale of $150,000.

Solution Summary: Theorem used: Separation of Variables: The solution of the differential equation is given implicitly by the equation displaystyle