Explanation Given Information: The department offers 3 courses, department may offer up to 45 sections total, up to 5000 students would like to a course and there are 60 professors, sections of ancient history have 100 students each, sections of medieval history have 50 students each and sections of modern history have 200 students each and modern history sections are taught by a team of 2 professors, while ancient and medieval history need only 1 professor per section and ancient history nets the university $ 10 , 000 per section, medieval history nets the university $ 20 , 000 per section and modern history nets the university $ 30 , 000 per section. Formula used: Step 1: Use slack variables and write the inequalities in equation form. Step 2: Write the equations in an augmented matrix. Step 3: Choose the smallest negative number on the left side of the bottom row and pivot the column. Step 4: Select the pivot which is the smallest of the test ratio a b , where a is the entry in the right most column and b is the corresponding entry in the pivot column. Step 5: Clear the pivot column using the pivot and label the pivot row with the label of the pivot column. Step 6: Repeat the above steps till all the entries in the bottom row are non negative. Use row operations to clear the column of the augmented matrix. Calculation: Consider department offers 3 courses, department may offer up to 45 sections total, up to 5000 students would like to a course and there are 60 professors, sections of ancient history have 100 students each, sections of medieval history have 50 students each and sections of modern history have 200 students each and modern history sections are taught by a team of 2 professors, while ancient and medieval history need only 1 professor per section and ancient history nets the university $ 10 , 000 per section, medieval history nets the university $ 20 , 000 per section and modern history nets the university $ 30 , 000 per section. Let x represent number of sections of ancient history, y represents the number of sections of medieval history and z represents the number of sections of modern history. Thus, the maximum profit is given by p = x + 2 y + 5 z . And, the constraints are x + y + z ≤ 45 10 x + 5 y + 20 z ≤ 500 x + y + 2 z ≤ 60 x ≥ 0 , y ≥ 0 , z ≥ 0 Consider the maximisation problem. Maximize p = x + 2 y + 5 z Subject to x + y + z ≤ 45 10 x + 5 y + 20 z ≤ 500 x + y + 2 z ≤ 60 x ≥ 0 , y ≥ 0 , z ≥ 0 Step 1: Use slack variables and write the inequalities in equation form. Add the slack variables s , t and u . x + y + z + s = 45 10 x + 5 y + 20 z + t = 500 x + y + 2 z + u = 60 Step 2: Write the equations in an augmented matrix. The equations are x + y + z + s = 45 10 x + 5 y + 20 z + t = 500 x + y + 2 z + u = 60 − x − 2 y − 3 z + p = 0 Write the equations in augmented matrix. x y z s t u p s 1 1 1 1 0 0 0 45 t 10 5 20 0 1 0 0 500 u 1 1 2 0 0 1 0 60 − 1 − 2 − 3 0 0 0 1 0 Step 3: Choose the smallest negative number on the left side of the bottom row and pivot the column. Since, the smallest negative number on the left side of the bottom row is − 3 . Thus, choose the left most entry and pivot the z column. Step 4: Select the pivot which is the smallest of the test ratios a b , where a is the entry in the right most column and b is the corresponding entry in the pivot column. Since, the test ratios are neither 0 nor negative. Thus, the test ratios are 45 1 = 45 , 500 20 ≈ 25 and 60 2 = 30 . Since, 25 is smallest. Thus, the pivot is 20 in column z . x y z s t u p s 1 1 1 1 0 0 0 45 t 10 5 20 0 1 0 0 500 u 1 1 2 0 0 1 0 60 − 1 − 2 − 3 0 0 0 1 0 Step 5: Clear the pivot column using the pivot and label the pivot row with the label of the pivot column. Since, the pivot is t in column z and replace t with z . Thus, z is the active variable. Use row operations to clear the pivot column of the augmented matrix. Use the row operations 20 R 1 − R 2 , 10 R 3 − R 2 and 20 R 4 + 3 R 2 . x y z s t u p s 10 15 0 20 − 1 0 0 400 z 10 5 20 0 1 0 0 500 u 0 5 0 0 − 1 10 0 100 10 − 25 0 0 0 0 20 1500 Step 6: Repeat the above steps till all the entries in the bottom row are non negative

Finite Mathematics and Applied Calculus (MindTap Course List)

7th Edition

ISBN: 9781337274203

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 6.3, Problem 25E

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To calculate: The number of sections of each course should the department offer to maximize profit, the value of maximum profit and the number of unused time slots, number of students who did not get into classes or any professor without anything to teach.

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Chapter 6.3, Problem 26E

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Finite Mathematics and Applied Calculus (MindTap Course List)

Ch. 6.1 - In Exercises 1-26, sketch the region that...Ch. 6.1 - In Exercises 1-26, sketch the region that...Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - In Exercises 1-26, sketch the region that...Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E

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The number of sections of each course should the department offer to maximize profit, the value of maximum profit and the number of unused time slots, number of students who did not get into classes or any professor without anything to teach.

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To calculate: The number of sections of each course should the department offer to maximize profit, the value of maximum profit and the number of unused time slots, number of students who did not get into classes or any professor without anything to teach.

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