Explanation Given information: f ( x ) = 0.01 x e − x / 10 , x ≥ 0 . The machine costs $ 60 , 000 and a manufacturer expects the machine to generate $ 5000 of additional income for every thousand hours of use. Explanation: A random variable X is thousand hours with a density function f ( x ) = 0.01 x e − x / 10 , x ≥ 0 . A manufacturer expects the machine to generate $ 5000 of additional income for every thousand hours of use, but the machine costs $ 60 , 000 . To determine whether the manufacturer should purchase the new equipment or not first find expected number of thousand hours of use. The formula to find the expected value is, E ( x ) = ∫ A B x f ( x ) d x Using this formula to find the expected value Thus, E ( x ) = ∫ 0 ∞ x 0.01 x e − x / 10 d x = 0.01 ∫ 0 ∞ x 2 e − x / 10 d x It is an indefinite integral, Thus, E ( x ) = 0.01 ⋅ lim b → ∞ ∫ 0 b x 2 e − x / 10 d x Use integration by parts rule to solve the integration ∫ 0 b x 2 e − x / 10 d x . Integration by parts rule is, ∫ f ( x ) g ( x ) = f ( x ) G ( x ) − ∫ f ' ( x ) G ( x ) d x . Thus here f ( x ) = x 2 and g ( x ) = e − x / 10 . By using by parts rule of integration, ⇒ ∫ x 2 e − x / 10 d x = x 2 . ( − 10 e − x / 10 ) − ∫ 2 x

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ISBN: 9780137590810

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

Publisher: PEARSON

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Chapter 12, Problem 20RE

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Whether the manufacturer should purchase the new equipment or not, if the machine costs $60,000 and a manufacturer expects the machine to generate $5000 of additional income for every thousand hours of use.

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Chapter 12, Problem 19RE

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Whether the manufacturer should purchase the new equipment or not, if the machine costs $ 60 , 000 and a manufacturer expects the machine to generate $ 5000 of additional income for every thousand hours of use.

Solution Summary: The author explains that a manufacturer expects the machine to generate 5000 of additional income for every thousand hours of use.

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Whether the manufacturer should purchase the new equipment or not, if the machine costs $60,000 and a manufacturer expects the machine to generate $5000 of additional income for every thousand hours of use.

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