Explanation Given Information: The three models the sprinkle, the storm and the hurricane requires the below mentioned amount of cloth, metal and wood to make it. Also, the table also shows the profit out of each model and total availability of the materials. Sprinkle Storm Hurricane Total available Cloth (square yards) 1 2 2 600 Metal (pounds) 2 1 3 600 Wood (pounds) 1 3 6 600 Profit ( $ ) 1 1 2 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 ratios 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: let the number of sprinkles, storms and hurricanes the society made be x , y , z respectively. Therefore, from the given table, The profit function can be sum of the product of number of each models made and profit from that model. Therefore, the profit function p can be p = x + y + 2 z The total availability of the material is a constraint. Therefore, the constraints can be derived from the amount of materials to be used in making a model and their availability. Therefore, considering the availability of cloth to be 600 square yards. x + 2 y + 2 z ≤ 600 Considering the availability of metal to be 600 pounds. 2 x + y + 3 z ≤ 600 Considering the availability of wood to be 600 pounds. x + 3 y + 6 z ≤ 600 Consider the constraints x + 2 y + 2 z ≤ 600 2 x + y + 3 z ≤ 600 x + 3 y + 6 z ≤ 600 x ≥ 0 , y ≥ 0 , z ≥ 0 Step 1: Use slack variables and write the inequalities in equation form. Add the slack variables s and t . x + 2 y + 2 z + s = 600 2 x + y + 3 z + t = 600 x + 3 y + 6 z + u = 600 Step 2: Write the equations in an augmented matrix. The equations are x + 2 y + 2 z + s = 600 2 x + y + 3 z + t = 600 x + 3 y + 6 z + u = 600 − x − y − 2 z + p = 0 Write the equations in augmented matrix. x y z s t u p s 1 2 2 1 0 0 0 600 t 2 1 3 0 1 0 0 600 u 1 3 6 0 0 1 0 600 p − 1 − 1 − 2 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 − 2 . Thus, 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 600 2 = 300 , 600 3 = 200 and 600 6 = 100 . Here smallest test ratio is 100. Thus, the pivot is 6 in column z . x y z s t u p s 1 2 2 1 0 0 0 600 t 2 1 3 0 1 0 0 600 u 1 3 6 0 0 1 0 600 p − 1 − 1 − 2 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 u in column z and replace u with z . Thus, z is the active variable. Use row operations to clear the pivot column of the augmented matrix. Use the row operations 3 R 1 − R 3 , 2 R 2 − R 3 and 3 R 4 + R 3 as, x y z s t u p s 2 3 0 3 0 − 1 0 1200 t 3 − 1 0 0 2 − 1 0 600 z 1 3 6 0 0 1 0 600 p − 2 0 0 0 0 1 3 600 Step 6: Repeat the above steps till all the entries in the bottom row are non-negative

Bundle: Finite Mathematics, 7th + WebAssign Printed Access Card for Waner/Costenoble's Finite Mathematics, 7th Edition, Single-Term

7th Edition

ISBN: 9781337604956

Author: Waner

Publisher: CENGAGE L

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Chapter 5, Problem 37RE

To determine

To calculate: The number of each types of model the society should male to increase its profits. The three models the sprinkle, the storm and the hurricane requires the below mentioned amount of cloth, metal and wood to make it. Also, the table also shows the profit out of each model and total availability of the materials.

Sprinkle Storm Hurricane Total available

Cloth (square yards) 1 2 2 600

Metal (pounds) 2 1 3 600

Wood (pounds) 1 3 6 600

Profit ($) 1 1 2

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Chapter 5, Problem 36RE

Chapter 5, Problem 38RE

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Bundle: Finite Mathematics, 7th + WebAssign Printed Access Card for Waner/Costenoble's Finite Mathematics, 7th Edition, Single-Term

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	Sprinkle	Storm	Hurricane	Total available
Cloth (square yards)	1	2	2	600
Metal (pounds)	2	1	3	600
Wood (pounds)	1	3	6	600
Profit ($)	1	1	2

Solution Summary: The author explains the formula used to calculate the number of sprinkles, storms and hurricanes the society should male to increase its profits.

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