Explanation Given: The initial amount invested is $100,000. Withdrawals are made at the rate of $20,000 per year. After 5 years the amount becomes $15,000. Formula used: If money is continuously withdrawn from the account at the rate of $ m per year, then A satisfies the equation d A d t − r A = − m ; where A ( 0 ) = P , d A d t is the rate of change of the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 withdrawals. 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 invested is $100,000. Withdrawals are made at the rate of $20,000 per year. After 5 years the amount becomes $15,000. Then substitute, m = 20 , 000 and P = 100 , 000 in the above equation. Therefore, the amount A at any time t must satisfy d A d t − r A = − 20 , 000 ; A ( 0 ) = 100 , 000 . Now find the integrating factor. I ( t ) = e ∫ − r d t = e − r t Multiply both sides of the equation by the integrating factor. e − r t d A d t − r e − r t A = − 20 , 000 e − r t ( A e − r t ) ′ = − 20 , 000 e − r t [ ∵ ( x y ) ′ = x y ′ + x ′ y ] Integrating on both sides, ∫ ( A e − r t ) ′ d t = ∫ − 20 , 000 e − r t d t A e − r t = − 20 , 000 ( e − r t − r ) + C A e − r t = 20 , 000 r e − r t + C A = 20 , 000 r + C e r t The initial condition is A ( 0 ) = 100 , 000

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Chapter 9.3, Problem 66E

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To find: The continuous compound rate of interest that has earned over 5-year period.

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Chapter 9.3, Problem 65E

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The continuous compound rate of interest that has earned over 5-year period.

Solution Summary: The author calculates the continuous compound rate of interest that has earned over a 5-year period. The integrating factor is given by I(x)=edisplaystyle

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To find: The continuous compound rate of interest that has earned over 5-year period.

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