Explanation Given information: $ 25 , 000 is the loan amount. The interest rate is 7 % which is compounded continuously. Daily payments are made totalling $ 4800 per year. Explanation: The two influences are such that the rate at which interest is added and the payments are made continuously into a savings account at the rate of $ 4800 per year which reduces loan. The rate of change is the net effect of these two influences. Thus, the first order linear differential equation is y ' = 0.07 y − 4800 . The initial amount of loan is $ 25 , 000 , so y ( 0 ) = 25 , 000 . Solve this differential equation using integrating factor as follows, Step 1: Write the differential equation y ' = 0.07 y − 4800 in standard form, ⇒ y ' − 0.07 y = − 4800 . Find integrating factor e A ( t ) . Compare y ' − 0.07 y = − 4800 and y ' + a ( t ) y = b ( t ) , ⇒ a ( t ) = − 0. 07 , b ( t ) = − 4800 An integrating factor of the differential equation y ' ( t ) + a ( t ) y = b ( t ) is e A ( t ) , where A ( t ) = ∫ a ( t ) d t . ⇒ A ( t ) = ∫ − 0.07 d t = − 0.07 t ⇒ e A ( t ) = e − 0.07 t Hence, the integrating factor is e A ( t ) = e − 0.07 t . Step 2: Multiply both sides of the differential equation y ' = 0.07 y − 4800 by the integrating factor e − 0.07 t . ⇒ y ' e − 0.07 t − 0.07 y e − 0.07 t = − 4800 e − 0.07 t ⇒ y ' e − 0.07 t + ( − 0.07 ) y e − 0.07 t = − 4800 e − 0.07 t By the product rule of derivative, ⇒ d d t [ e − 0.07 t y ] = − 4800 e − 0.07 t Step 3: Integrating both sides and solving for y , ⇒ d d t [ e − 0.07 t y ] = ∫ − 4800 e − 0.07 t d t ⇒ e − 0.07 t y = − 4800 ( e − 0.07 t − 0.07 ) + C ⇒ e − 0.07 t y = ( 4800 0.07 ) e − 0

Calculus & Its Applications (14th Edition)

14th Edition

ISBN: 9780134437774

Author: Larry J. Goldstein, David C. Lay, David I. Schneider, Nakhle H. Asmar

Publisher: PEARSON

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Chapter 10.4, Problem 2E

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The time t required to pay off the loan of $ 25,000 taken for a new car. The interest rate is 7% compounded continuously, and the daily payments totalling $ 4800 per year. P(t) denotes the amount owed on the loan at time t.

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Chapter 10.4, Problem 1E

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Calculus & Its Applications (14th Edition)

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The time t required to pay off the loan of $ 25 , 000 taken for a new car. The interest rate is 7 % compounded continuously, and the daily payments totalling $ 4800 per year. P ( t ) denotes the amount owed on the loan at time t .

Solution Summary: The author calculates the time required to repay the loan of 25,000 for a new car. The rate of change is the net effect of these two influences.

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The time t required to pay off the loan of $ 25,000 taken for a new car. The interest rate is 7% compounded continuously, and the daily payments totalling $ 4800 per year. P(t) denotes the amount owed on the loan at time t.

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