Explanation Given Information: The provided information is as follows, “Two types of furniture- chairs and sofa is made by a manufacturer. Manufacture of chair requires 6 hours of carpentry, 2 hours of upholstery, and 1 hour of finishing where manufacture of sofa requires 3 hours of carpentry, 6 hours of upholstery, and 1 hour of finishing. The factory has 96 labor-hours for carpentry, 72 labor-hours of upholstery and 18 labor-hours of finishing and the profit per chair is $150 and for sofa $70.” Formula Used: To find the maximum and minimum value of objective function follow the given steps: 1)Translate the linear programming problem into mathematical language and graph the feasible solution set. 3) Determine the vertices of the feasible solution set obtained. 4) Find the value of objective function at each vertex and determine the feasible set. 5) Interpret the result. Addition property: For any real numbers a, b and c, if a = c , then a + b = c + b . Multiplication property: For any real numbers a, b and c ≠ 0 , if a = c , then a ⋅ b = c ⋅ b . Calculation: Make a table according to the given condition as: Chair Sofa Available time Carpenter 6 hours 3 hours At most 96 hours Finishing 1 hour 1 hours at most 18 hours Upholstery 2 hours 6 hours at most 72 hours Profit $150 $70 Let number of chairs be x and number of sofas be y. Then total profit is 150 x + 70 y From the table, For carpentry, the inequality is: 6 x + 3 y ≤ 96 For finishing, the inequality is: x + y ≤ 18 For upholstery, the inequality is: 2 x + 6 y ≤ 72 The inequalities in standard form are: y ≤ − 2 x + 32 y ≤ − x + 18 y ≤ − 1 3 x + 12 x ≥ 0 , y ≥ 0 Assume the inequality as equation to find coordinate pairs. Now to find the intersection point of y = − 2 x + 32 and y = − x + 18 , equate the values of y , − 2 x + 32 = − x + 18 Apply the addition principle and add x on both sides of the equation. − 2 x + 32 = − x + 18 − 2 x + 32 + x = − x + x + 18 − x + 32 = 18 Apply the addition principle and add − 32 on both sides of the equation. − x + 32 = 18 − x + 32 + ( − 32 ) = 18 + ( − 32 ) − x = − 14 x = 14 Put this value x = 14 in y = − x + 18 to get, y = − x + 18 y = − 14 + 18 y = 4 The intersection point is ( 14 , 4 ) . Now to find the intersection of y = − 1 3 x + 12 and y = − x + 18 equate the values of y , − 1 3 x + 12 = − x + 18 Apply the addition principle and add x on both sides of the equation. − 1 3 x + 12 = − x + 18 − 1 3 x + 12 + x = − x + x + 18 − x + 3 x 3 + 12 = 18 2 x 3 + 12 = 18 Apply the addition principle and add − 12 on both sides of the equation

12th Edition

ISBN: 9781323788707

Author: Goldstein

Publisher: PEARSON

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Chapter 3.3, Problem 23E

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To calculate: The maximum profit if the profit per chair is $150 and for sofa $70.

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Chapter 3.3, Problem 22E

Chapter 3.3, Problem 24E

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Chapter 3 Solutions

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The maximum profit if the profit per chair is $150 and for sofa $70.

Solution Summary: The author explains the formula used to calculate the maximum and minimum value of objective function.

To calculate: The maximum profit if the profit per chair is $150 and for sofa $70.

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Chapter 3 Solutions