Explanation Formula used: If money is continuously withdrawn from the account at the rate of $ m per year, then A satisfies the equation d A d t − r A = − m ; where A ( 0 ) = P where d A d t is the rate of change of the amount A , rA is the rate of growth from continuous compounding and m is the rate of withdrawal. Integrating factor: The integrating factor of the First-Order linear differential equation y ′ + f ( x ) y = g ( x ) is I ( x ) = e ∫ f ( x ) d x . Calculation: It is given that the initial amount deposited is $60,000 into an account at a rate of 5% which is compounded continuously and then money is continuously drawn at a constant rate of $5,000 a year until the account is depleted. Then, substitute r = 0.05 , m = 5 , 000 and P = 60 , 000 in the above equation. Therefore, the amount A at any time t must satisfy the equation d A d t − 0.05 A = − 5 , 000 ; A ( 0 ) = 60 , 000 . Now find the integrating factor. I ( t ) = e ∫ − 0.05 d t = e − 0.05 t Multiply both sides of the standard form by the integrating factor. e − 0.05 t d A d t − 0.05 e − 0.05 t A = − 5 , 000 e − 0.05 t ( A e − 0.05 t ) ′ = − 5 , 000 e − 0.05 t [ ∵ ( x y ) ′ = x y ′ + x ′ y ] Integrating on both sides, ∫ ( A e − 0.05 t ) ′ d t = ∫ − 5 , 000 e − 0

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Chapter 9, Problem 43RE

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To find: The amount in the account at time t, the value of t at which the amount becomes $0 and the total amount withdrawn from the account.

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Chapter 9, Problem 42RE

Chapter 9, Problem 44RE

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Chapter 9 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

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The amount in the account at time t , the value of t at which the amount becomes $0 and the total amount withdrawn from the account.

Solution Summary: The author calculates the integrating factor of the First-Order linear differential equation.

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To find: The amount in the account at time t, the value of t at which the amount becomes $0 and the total amount withdrawn from the account.

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Chapter 9 Solutions