(A) Explanation Given: The continuous random variable X with probability density function is f ( x ) = { 3 x 25 − x 2 125 if 0 ≤ x ≤ 5 0 otherwise , for the weekly demanded magazine (in thousands of copies). Property used: Cumulative distribution functions : If f is a probability function and F ( x ) = ∫ − ∞ x f ( t ) d t is the associated cumulative distribution function, then 1. F ′ ( x ) = f ( x ) wherever f is continuous. 2. 0 ≤ F ( x ) ≤ 1 , − ∞ < x < ∞ . 3. F ( x ) is non-decreasing on ( − ∞ , ∞ ) . Calculation: Find the probability that the number of magazine ordered by the distributor exceeds 3,000 copies by integrating the given function with 3 ≤ x ≤ 5 . P ( 3 ≤ x ≤ 5 ) = ∫ 3 5 f ( x ) d x = ∫ 2 5 ( 3 x 25 − x 2 125 ) d x = 0.024 ∫ 2 5 ( x 25 − x 2 ) d x That is, P ( 3 ≤ x ≤ 5 ) = 0.024 ∫ 2 5 ( x 25 − x 2 ) d x (1) The term x 25 − x 2 can be written as x ( 25 − x 2 ) 1 2 . Find the indefinite integral of ∫ ( x 25 − x 2 ) d x . Let the variable v be 25 − x 2 . Differentiate v with respect to x . d d x ( v ) = d d x ( 25 − x 2 ) d v d x = 0 − 2 x d v = − 2 x d x x d x = − d v 2 Substitute v = 25 − x 2 and x d x = − d v 2 in the integral ∫ ( x 25 − x 2 ) d x , ∫ ( x 25 − x 2 ) d x = ∫ − d v 2 v ( 1 2 ) = − 1 2 ∫ v ( 1 2 ) d v That is, ∫ ( x 25 − x 2 ) d x = − 1 2 ∫ v ( 1 2 ) d v (2) Integrate ∫ v ( 1 2 ) d v (B) To determine To find: The probability when the demand of the magazine does not exceed 4,000 copies. (C) To determine To find: The number of magazine copies to be ordered so that the probability does not exceed 0.95.

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Chapter 11.2, Problem 60E

(A)

To determine

To find: The probability that the number of magazine ordered by the distributor is more than 3,000 copies.

(B)

To determine

To find: The probability when the demand of the magazine does not exceed 4,000 copies.

(C)

To determine

To find: The number of magazine copies to be ordered so that the probability does not exceed 0.95.

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Chapter 11.2, Problem 59E

Chapter 11.2, Problem 61E

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Chapter 11 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

Ch. 11.1 - Evaluate the following, if it converges: 3dx(x1)2.Ch. 11.1 - Prob. 2MP Ch. 11.1 - Prob. 3MP Ch. 11.1 - Prob. 4MP Ch. 11.1 - Prob. 5MP Ch. 11.1 - Prob. 6MP Ch. 11.1 - Prob. 1ED Ch. 11.1 - Prob. 2ED Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E

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The probability that the number of magazine ordered by the distributor is more than 3,000 copies.

Solution Summary: The author calculates the probability that the number of magazine ordered by the distributor exceeds 3,000 copies by integrating the given function with 3le x

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To find: The probability that the number of magazine ordered by the distributor is more than 3,000 copies.

To find: The probability when the demand of the magazine does not exceed 4,000 copies.

To find: The number of magazine copies to be ordered so that the probability does not exceed 0.95.

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Chapter 11 Solutions