a. Explanation Given: The two types of chairs are Standard and Deluxe chairs. Each standard chair needs 4 hours to construct and finish and each Deluxe chair needs 6 hours to construct and finish. Upholstering for a standard chair and Deluxe chair requires 2 hours and 6 hours, respectively. Available hours per day for construction and upholstering are 480 hours and 300 hours, respectively. The revenue of each standard chair and each Deluxe chair is $178 and $267 respectively. Calculation: Let x denote the number of Standard chairs and y denote the number of Deluxe chairs. The total hours needed to construct and finish the Standard chairs is 4 x and for Deluxe chairs is 6 y . The total available hours each day for construction is 480 hours. Note that the total hours cannot exceed 480 hours which can be written as the inequality, 4 x + 6 y ≤ 480 . The total available hours per day for upholstering are 300 hours. The total hours cannot exceed 300 hours which can be written as the inequality, 2 x + 6 y ≤ 300 . In the context of this application, the number of hours cannot be negative. Therefore, it can be concluded that the value of x and y must be nonnegative. That is, x ≥ 0 and y ≥ 0 . Thus, the inequalities representing manufacturing is, { 4 x + 6 y ≤ 480 2 x + 6 y ≤ 300 x ≥ 0 , y ≥ 0 (1) From the given information, since the revenue of each standard chair and each Deluxe chair is $178 and $267 respectively, the objective function to be maximized is f = 178 x + 267 y . Sketch the graph of the inequalities (1) and identify the corners of the region. The boundary equations of the above inequalities are, { 4 x + 6 y = 480 2 x + 6 y = 300 x = 0 , y = 0 (2) Sketch the graph of the straight lines 4 x + 6 y = 480 , 2 x + 6 y = 300 , x = 0 and y = 0 as shown below in Figure 1. From Figure 1, it is observed that the equation (2) divides the plane into 4 parts. The solution region of the given inequality set is obtained as follows. Take any arbitrary test point in region I, to test the solution region of the equation (1). Let ( 160 , 0 ) be the test point of the region I. Substitute x = 160 and y = 0 in equation (1), { 4 x + 6 y ≤ 480 2 x + 6 y ≤ 300 x ≥ 0 , y ≥ 0 = { 4 ( 160 ) + 6 ( 0 ) ≤ 480 2 ( 160 ) + 6 ( 0 ) ≤ 300 160 ≥ 0 , 0 ≥ 0 That is, the above simplified inequality is { 640 ≤ 480 320 ≥ 300 160 ≥ 0 , 0 ≥ 0 . Since the expression 640 ≤ 480 and 320 ≥ 300 is not true, the point ( 160 , 0 ) does not satisfy the equation (1). Thus, the region I is not the solution region of the equation (1). Take any arbitrary test point in the region II, to check the solution region of the equation (1). Let ( 40 , 60 ) be the test point of the region II. Substitute x = 40 and y = 60 in the equation (1), { 4 x + 6 y ≤ 480 2 x + 6 y ≤ 300 x ≥ 0 , y ≥ 0 = { 4 ( 40 ) + 6 ( 60 ) ≤ 480 2 ( 40 ) + 6 ( 60 ) ≤ 300 40 ≥ 0 , 60 ≥ 0 That is, the above simplified inequality is { 520 ≤ 480 440 ≥ 300 40 ≥ 0 , 60 ≥ 0 . Since the expression 520 ≤ 480 and 440 ≥ 300 are not true, the point ( 40 , 60 ) does not satisfy the equation (1). Thus, the region II is not a solution region of the equation (1). Take any arbitrary test point in the region III to test the solution region of the equation (1). Let ( 130 , 0 ) be the test point of the region III. Substitute x = 130 and y = 0 in the equation (1), { 4 x + 6 y ≤ 480 2 x + 6 y ≤ 300 x ≥ 0 , y ≥ 0 = { 4 ( 130 ) + 6 ( 0 ) ≤ 480 2 ( 130 ) + 6 ( 0 ) ≤ 300 130 ≥ 0 , 0 ≥ 0 That is, the above simplified inequality is { 520 ≤ 480 360 ≥ 300 130 ≥ 0 , 0 ≥ 0 Since the expression 520 ≤ 480 and 360 ≥ 300 are not true, the point ( 40 , 60 ) does not satisfy the equation (1) b. To determine To find: The number of chairs produced each day to maximize the daily revenue.

College Algebra in Context with Applications for the Managerial, Life, and Social Sciences (5th Edition)

5th Edition

ISBN: 9780134179025

Author: Ronald J. Harshbarger, Lisa S. Yocco

Publisher: PEARSON

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Chapter 8.2, Problem 28E

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To find: The maximum possible daily revenue.

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To find: The number of chairs produced each day to maximize the daily revenue.

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Chapter 8.2, Problem 27E

Chapter 8.2, Problem 29E

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Chapter 8 Solutions

College Algebra in Context with Applications for the Managerial, Life, and Social Sciences (5th Edition)

Ch. 8.1 - In Exercises 16, graph each inequality. 1. y 5x ...Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E Ch. 8.1 - In Exercises 710, match each solution of a system...Ch. 8.1 - Prob. 10E

Ch. 8.1 - Prob. 11E Ch. 8.1 - In Exercises 1116, the graph of the boundary...Ch. 8.1 - Prob. 13E Ch. 8.1 - Prob. 14E Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - Prob. 17E Ch. 8.1 - Prob. 18E Ch. 8.1 - For each system of inequalities in Exercises 1722,...Ch. 8.1 - Prob. 20E Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - Prob. 23E Ch. 8.1 - In Exercises 2326, graph the solution of the...Ch. 8.1 - Prob. 25E Ch. 8.1 - In Exercises 2326, graph the solution of the...Ch. 8.1 - Prob. 27E Ch. 8.1 - Prob. 28E Ch. 8.1 - Prob. 29E Ch. 8.1 - Sales Trix Auto Sales sells used cars. 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The maximum possible daily revenue.

Solution Summary: The author calculates the maximum possible daily revenue for Standard and Deluxe chairs, which is 21,360. The total hours needed for construction and upholstering are 480 hours and 300 hours respectively.

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To find: The maximum possible daily revenue.

To find: The number of chairs produced each day to maximize the daily revenue.

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Chapter 8 Solutions