Explanation Given: The initial sales of a company were $0 million. The sales were $2 million during first year and $5 million during third year. The annual sales grow 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 is given by the equation S ( t ) = M ( 1 − e − k t ) , M > 0 , k > 0 . The sales were $2 million during first year and $5 million during third year. That is, S ( 1 ) = 2 and S ( 3 ) = 5 . 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 = 5 in S ( t ) = 2 1 − e − k ( 1 − e − k t ) . S ( t ) = 2 1 − e − k ( 1 − e − k t ) 5 = 2 1 − e − k ( 1 − e − 3 k ) 5 ( 1 − e − k ) = 2 ( 1 − e − 3 k ) 5 − 5 e − k = 2 − 2 e − 3 k Simplify further as, 2 e − 3 k − 5 e − k + 3 = 0 . Consider t = e − k , then k = − ln t . Thus, the obtained equation becomes, 2 t 3 − 5 t + 3 = 0 . Factorize the cubic polynomial 2 t 3 − 5 t + 3 = 0 . By the trial and error method, suppose t = 1 . 2 ( 1 ) 3 − 5 ( 1 ) + 3 = 2 − 5 + 3 = 0 Thus, t − 1 is a factor of 2 t 3 − 5 t + 3 = 0

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Chapter 9.2, Problem 69E

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To find: The approximate value of k round off to one decimal place and M round off to the nearest million.

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Chapter 9.2, Problem 68E

Chapter 9.2, Problem 70E

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Chapter 9 Solutions

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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 value of k round off to one decimal place and M round off to the nearest million.

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