Explanation Given: The initial amount deposited is $10,000. The interest is compounded continuously at a rate of 6.25%. And then money is continuously deposited at a constant rate of $5,000 a year. Formula used: If money is continuously deposited into 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 , 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 deposit. Integrating factor: The integrating factor of a function f ( x ) is given by I ( x ) = e ∫ f ( x ) d x . Calculation: Given that the initial amount deposited is $10,000 into an account at a rate of 6.25% which is compounded continuously. And then money is continuously drawn at a constant rate of $5,000 a year. Then substitute r = 0.0625 , m = 5 , 000 and P = 10 , 000 in the above equation. Therefore, the amount A at any time t must satisfy d A d t − 0.0625 A = 5 , 000 ; A ( 0 ) = 10 , 000 . Now, find the integrating factor. I ( t ) = e ∫ − 0.0625 d t = e − 0.0625 t Multiply both sides of the standard form by the integrating factor. e − 0.0625 t d A d t − 0.0625 e − 0.0625 t A = 5 , 000 e − 0.0625 t ( A e − 0.0625 t ) ′ = 5 , 000 e − 0.0625 t [ ∵ ( x y ) ′ = x y ′ + x ′ y ] Integrating on both sides, ( A 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 9.3, Problem 64E

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To find: The amount in the account at time t, and the amount in the account after 10 years.

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Chapter 9.3, Problem 63E

Chapter 9.3, Problem 65E

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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 , and the amount in the account after 10 years.

Solution Summary: The author calculates the integrating factor of a function f(x) by multiplying both sides of the standard form.

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To find: The amount in the account at time t, and the amount in the account after 10 years.

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Chapter 9 Solutions