Explanation Given information: The 72 new books each year, which are elementary, intermediate, and advanced. The textbook publisher has policy for the new books is to publish at least four advanced books at least three times as many elementary books as intermediate books for least twice as many intermediate books as advanced books. On an average the annual profits on each types of books are: $8000 for each elementary book, $7000 for each intermediate book and $1000 for each advanced book. Formula used: For the inequality y ≥ m x + c or y ≤ m x + c , Draw the graph of y = m x + c . The graph of y ≥ m x + c consists of all points above or on the line, and y ≤ m x + c consists of all points below or on the line. Calculation: Consider the number of elementary books, intermediate books and advanced books are x , y and z respectively. As, he publishes 72 books each year, so x + y + z = 72 z = 72 − x − y As, the company publishes at least four advanced books, so z ≥ 4 As from above, z = 72 − x − y . z ≥ 4 72 − x − y ≥ 4 y ≤ − x + 68 Also, the company publishes at least three times as many elementary as intermediate books, so x ≥ 3 y y ≤ 1 3 x And, the company publishes at least three twice as many intermediate books as advanced books, so y ≥ 2 z y ≥ 2 ( 72 − x − y ) y ≤ − 2 3 x + 48 The annual profits on each types of books are: $8000 for each elementary book, $7000 for each intermediate book and $1000 for each advanced book So, the objective function which has to maximize is: M = 8000 x + 7000 y + 1000 z M = 8000 x + 7000 y + 1000 ( 72 − x − y ) M = 72000 + 7000 x + 6000 y So, the objective function is: Maximize: M = 72000 + 7000 x + 6000 y As, the number of resources cannot be negative, so x ≥ 0 , y ≥ 0 and z ≥ 0 . Therefore, constraints equations are: { y ≤ − x + 68 y ≤ 1 3 x y ≤ − 2 3 x + 48 x ≥ 0 , y ≥ 0 From above, there are five linear equations, write those equations in the form of y ≤ m x + c or y ≥ m x + c , so For the 1 st inequality, The intercept on x -axis is at ( 68 , 0 ) and y intercept is at ( 0 , 68 )

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ISBN: 9781323788707

Author: Goldstein

Publisher: PEARSON

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Chapter 3, Problem 8RE

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To calculate: The number of elementary books for the company on which the the average, the annual profits on each types of books are: $8000 for each elementary book, $7000 for each intermediate book and $1000 for each advanced book and the textbook publisher has policy for the new books is to publish at least four advanced books at least three times as many elementary books as intermediate books for least twice as many intermediate books as advanced book and publisher produced 72 new books each year, which are elementary, intermediate, and advanced.

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Solution Summary: The author calculates the annual profits on elementary, intermediate, and advanced books for a company that publishes 72 new books each year.

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