Explanation Given Information: The investment is compounded continuously, the investment grows at a rate that is proportional to the amount in the amount in the account. d P d t = k P Where P is in dollars, t is in the years, and k is a constant. If $ 20 , 000 is invested and the amount in the account after 22 years is $ 280 , 264 . Formula used: Separable equation is of the form: g ( y ) d y = f ( x ) d x Integration on both the sides yields the solution of this equation. Calculation: Consider the investment is compounded continuously, the investment grows at a rate that is proportional to the amount in the amount in the account. d P d t = k P Where P is in dollars, t is in the years, and k is a constant. If $ 20 , 000 is invested and the amount in the account after 22 years is $ 280 , 264 . Firstly, separate the above differential equation as: d P P = k d t Integrate the provided function as: ∫ d P P = ∫ k d t ln P = k t + C P = e k t + C P = C e k t At the given condition t = 0 , P = 20 , 000 put in the above equation as: 20 , 000 = C e e ( 0 ) C = 20 , 000 At the given condition t = 22 , P = 280 , 264 put in the above equation as: 280 , 264 = 20 , 000 e 22 t 14

Mathematical Applications for the Management, Life, and Social Sciences

11th Edition

ISBN: 9781305108042

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Differential Equations

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Chapter 12.5, Problem 42E

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To calculate: The investment function after time t, and also interest rate of the investment if the account rate is dPdt=kP and $20,000 is invested and the amount in the account after 22 years is $280,264.

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Chapter 12.5, Problem 41E

Chapter 12.5, Problem 43E

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Mathematical Applications for the Management, Life, and Social Sciences

Ch. 12.1 - 1. True or false: (a) (b) Ch. 12.1 - Prob. 2CP Ch. 12.1 - 1. If what is Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E Ch. 12.1 - Prob. 5E Ch. 12.1 - Evaluate the integrals in Problems 5-28. Check...Ch. 12.1 - Prob. 7E Ch. 12.1 - Evaluate the integrals in Problems 5-28. Check...

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The investment function after time t , and also interest rate of the investment if the account rate is d P d t = k P and $ 20 , 000 is invested and the amount in the account after 22 years is $ 280 , 264 .

Solution Summary: The author calculates the investment function after time t and interest rate if the account rate is P=20,000e0.12t.

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To calculate: The investment function after time t, and also interest rate of the investment if the account rate is dPdt=kP and $20,000 is invested and the amount in the account after 22 years is $280,264.

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Chapter 12 Solutions