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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

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Chapter
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BuyFindarrow_forward

Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

APPLICATIONS

Manufacturing Furniture A furniture factory makes wooden tables, chairs, and armoires. Each piece of furniture requires three operation: cutting the wood, assembling, and finishing. Each operation requires the number of hours given in the table. The workers in the factory can provide 300 h of cutting. 400 h or assembling, and 590 h of finishing each work week. How many tables, chairs, and armoires should he produced so that all available labor-hours are used? Or is this impossible?

Table Chair Armoire
Cutting (h) 1 2 1 1
Assembling (h) 1 2 1 1 2 1
Finishing (h) 1 1 1 2 2

To determine

To find:

The number of tables, chair, and armoire required so that all available labor hours are used.

Explanation

Given:

A furniture factory makes wooden tables, chairs, and armoires. Each piece of furniture requires three operations: cutting the wood, assembling, and finishing. Each operation requires the number of hours given in the table. The workers in the factory can provide 300 h of cutting. 400 h or assembling, and 590 h of finishing each work week.

Table Chair Armoire
Cutting (h) 12 1 1
Assembling (h) 12 112 1
Finishing (h) 1 112 2

Table (1)

Approach:

Let x, y, and z represents the number of table, chairs, and armoires.

Write the augmented matrix of the system.

Use elementary row operations to change the augmented matrix to row-echelon form.

Write the new system of equations that corresponds to the row-echelon form of the augmented matrix and solve by back-substitution.

Calculation:

The system of equations is given below.

x2+y+z=300x2+32y+z=400x+32y+2z=590

Consider the system of equations,

{x2+y+z=300x2+32y+z=400x+32y+2z=590

Or,

{x+2y+2z=600x+3y+2z=8002x+3y+4z=1180

The augmented matrix is given by,

[1226001328002341180]

Transform the augmented matrix into row echelon form.

Perform the transformation R2R2R1.

[1226001132228006002341180][1226000102002341180]

Perform the transformation R3R32R1

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