(A) Explanation Given: The growth rate of annual sales S is proportional to the difference between the sales and a $20 million. The initial sales are $0 million and $3 million for the second year of operation. 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 Calculation: From the given condition, it is observed that the annual sales follow a growth model proportional to the difference between the sales and a $20 million. Thus, a limited growth model will suit the problem. Let S denote the volume of a mothball after t weeks. Then, by the limited growth model, d S d t = k ( 20 − S ) The initial sale is given as $0 million. Thus, S ( 0 ) = 0 . From the condition that the initial sale is $3 million for the second year, S ( 2 ) = 3 The first step is to use separation of variables to find the general solution to the differential equation d S d t = k ( 20 − S ) . d S = k d t ( 20 − S ) d S ( 20 − S ) = k d t ∫ 1 ( 20 − S ) d S = ∫ k d t Obtain the integrals on both side as, − ln ( 20 − S ) = k t + C ln ( 20 − S ) = − k t − C ( 20 − S ) = e − k t − C ( 20 − S ) = e − C e − k t Simplify further as, ( 20 − S ) = A e − k t , where A = e − C S = 20 − A e − k t That is, S ( t ) = 20 − A e − k t (B) To determine To find: The values for S ( 10 ) and S ′ ( 10 ) S and to interpret the conclusion. (C) To determine To find: The year at which the sales is expected to reach $15 million.

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 4MP

(A)

To determine

To find: The sales S during the year t.

(B)

To determine

To find: The values for S(10) and S′(10) S and to interpret the conclusion.

(C)

To determine

To find: The year at which the sales is expected to reach $15 million.

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

Ch. 9.1 - Show that y=Cx+1 is the general solution of the...Ch. 9.1 - Prob. 2MP Ch. 9.1 - Prob. 3MP Ch. 9.1 - Prob. 4MP Ch. 9.1 - Prob. 5MP Ch. 9.1 - Prob. 1ED Ch. 9.1 - Prob. 2ED Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E

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The sales S during the year t .

Solution Summary: The author analyzes how the growth rate of annual sales S is proportional to the difference between the sales and a 20 million. The initial sales are 0 million and 3 million for the second year.

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To find: The sales S during the year t.

To find: The values for S(10) and S′(10) S and to interpret the conclusion.

To find: The year at which the sales is expected to reach $15 million.

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Chapter 9 Solutions