Explanation Given: A new refrigerator costs $1,400. The value of the refrigerator depreciates to $100 over a time period of 20 years. The rate of change of the value is proportional to the value. Theorem used: Separation of Variables: The solution of the differential equation f ( y ) y ′ = g ( x ) is given implicitly by the equation ∫ f ( y ) d y = ∫ g ( x ) d x . Formula used: Unlimited growth model: If the rate of change with respect to time t of a quantity y is proportional to the amount present, then y satisfies the differential equation, d y d t = k y , k > 0 , t > 0 Calculation: Let y represent the value of the refrigerator. The rate of change of the value is proportional to the value. That is, the model used is unlimited growth which has the formula, d y d t = k y , k > 0 , t > 0 Rewrite the differential equation to the form of equation f ( y ) y ′ = g ( x ) as follows. d y y = k d t 1 y d y = k d t Compare the equation 1 y d y = k d t with equation f ( y ) y ′ = g ( x ) . It is observed that, f ( y ) = 1 y g ( t ) = k By the above mentioned theorem, the solution is given by, ∫ f ( y ) d y = ∫ g ( t ) d t ∫ 1 y d y = ∫ k d t ln y = k t + C 1 y = e k t + C 1 Simplify further as, y = e k t e C 1 = C e k t , where C = e C 1 The initial value of the refrigerator is $1,400

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

15th Edition

ISBN: 9781323047620

Author: Barnett, Ziegler, Byleen

Publisher: Pearson Education

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Chapter 9, Problem 40RE

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The value of the refrigerator after 5 years of purchase.

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Chapter 9, Problem 39RE

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Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

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The value of the refrigerator after 5 years of purchase.

Solution Summary: The author explains how the value of the refrigerator after 5 years of purchase is 723.59.

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