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: Unlimited growth model: d y d t = k y , k > 0 , t > 0 Calculation: The total corporate profits before tax were $1.84 trillion in 2010 and $ 2.14 trillion in 2015. Let y represents the total corporate profits. The rate at which the profit increases or decreases will be proportional to the total corporate profits. That is, d y d t = k y , k > 0 , t > 0 . Thus, the problem follows an unlimited growth model. Rewrite the differential equation d y d t = k y as, d y d t = k y d y y = k d t 1 y d y = k d t Comparing the equation 1 y d y = k d t and f ( y ) y ′ = g ( x ) , it is observed that f ( y ) = 1 y and g ( t ) = k . By the above theorem, the solution is given by, ∫ f ( y ) d y = ∫ g ( t ) d t ∫ 1 y d y = ∫ k d t ∫ ( 1 y ) d y = ∫ k d t ln y = k t + C 1 Simplify further as, e ln ( y ) = e k t + C 1 y = e k t + C 1 y = C e k t , where C = e C 1 The total corporate profits before tax were $1.84 trillion in 2010 and $ 2.14 trillion in 2015. The initial year is taken as 2010. That is, y ( 0 ) = 1.84 . Substitute t = 0 and y = 1.84 in y = C e k t , y = C e k t 1.84 = C e k ( 0 ) C = 1

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 66E

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Chapter 9.2, Problem 65E

Chapter 9.2, Problem 67E

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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 total corporate profits and the rate of change of total corporate profits in 2025.

Solution Summary: The author calculates the total corporate profits and rate of change in 2025. The solution of the differential equation is given implicitly by the equation.

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