Explanation Given: The initial sales of a company was $0 million. The sales was $2 million during first year and $4 million during third year. The annual sales grows at a rate proportional to the difference between the sales and a limit M . 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: The sales in million dollars is given by the equation, S ( t ) = M ( 1 − e − k t ) , M > 0 , k > 0 . Given S ( 1 ) = 2 and S ( 3 ) = 4 . Substitute t = 1 and S = 2 in S ( t ) = M ( 1 − e − k t ) . 2 = M ( 1 − e − k ) M = 2 1 − e − k Substitute M = 2 1 − e − k in S ( t ) = M ( 1 − e − k t ) , S ( t ) = 2 1 − e − k ( 1 − e − k t ) . Substitute t = 3 and S = 4 in S ( t ) = 2 1 − e − k ( 1 − e − k t ) . 4 = 2 1 − e − k ( 1 − e − 3 k ) 4 ( 1 − e − k ) = 2 ( 1 − e − 3 k ) 4 − 4 e − k = 2 − 2 e − 3 k Simplify further as, 2 e − 3 k − 4 e − k + 2 = 0 Let t = e − k . Then, k = − ln t . Then, 2 t 3 − 4 t + 2 = 0 . Factorize the cubic polynomial 2 t 3 − 4 t + 2 = 0 . By trial and error method, suppose t = 1 . 2 ( 1 ) 3 − 4 ( 1 ) + 2 = 2 − 4 + 2 = 0 Thus, t − 1 is a factor of 2 t 3 − 4 t + 2 = 0 . Divide the polynomial 2 t 3 − 4 t + 2 = 0 by t − 1 as follows

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ISBN: 9781323520192

Author: Pearson

Publisher: PEARSON C

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Differential Equations

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Chapter 9.2, Problem 70E

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To find: The approximate values of k to one decimal and M to the nearest million.

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Chapter 9.2, Problem 69E

Chapter 9.2, Problem 71E

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FIN 108 ISU LOOSE >IP<

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 approximate values of k to one decimal and M to the nearest million.

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