A manufacturer of women's blouses makes three types of blouses: sleeveless, short-sleeve, and long-sleeve. The time (in minutes) required by each department to produce a dozen blouses of each type is shown in the following table. SleevelessShort-SleeveLong-SleeveCutting  91215Sewing222428Packaging  6  8  8The cutting, sewing, and packaging departments have available a maximum of 84, 166, and 50 labor-hours, respectively, per day. How many dozens of each type of blouse can be produced each day if the plant is operated at full capacity?

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Asked Sep 15, 2019
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A manufacturer of women's blouses makes three types of blouses: sleeveless, short-sleeve, and long-sleeve. The time (in minutes) required by each department to produce a dozen blouses of each type is shown in the following table.

  Sleeveless Short-
Sleeve
Long-
Sleeve
Cutting   9 12 15
Sewing 22 24 28
Packaging   6   8   8

The cutting, sewing, and packaging departments have available a maximum of 84, 166, and 50 labor-hours, respectively, per day. How many dozens of each type of blouse can be produced each day if the plant is operated at full capacity?

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Expert Answer

Step 1

Let,

Number of sleeveless blouses manufactured per day (in dozen) = x

Number of short-sleeve blouses manufactured per day (in dozen) = y

Number of sleeveless blouses manufactured per day (in dozen) = z

Sleeveless
Short-sleeve
Long-sleeve
Cutting
Sewing
Packaging
9
12
15
22
24
28
6
8
8
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Sleeveless Short-sleeve Long-sleeve Cutting Sewing Packaging 9 12 15 22 24 28 6 8 8

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Step 2

So, as per the information available in the question, we can write these equations

9x 12y15z
5040
84 х 60
22x 24y
28z = 166 x 60
9960
50 x 60 3000
бх + 8y + 82
Above system of equations
can be represented in matrix form as,
12 151 x
[50401
9960
[3000
9
22
24 28y
8
Z
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9x 12y15z 5040 84 х 60 22x 24y 28z = 166 x 60 9960 50 x 60 3000 бх + 8y + 82 Above system of equations can be represented in matrix form as, 12 151 x [50401 9960 [3000 9 22 24 28y 8 Z

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Step 3

We will use Cramers rule to solv...

12 15
|22 24 28
9
The determinant of coefficient matrix, D
=
8
6
9(24 x 8 - 28 x 8) - 12(22 x 8 - 28 x 6) 15(22 x 8 - 28 x 6)
= 96
Similarly
15
|5040
12
24 28
9960
5760
Dr
3000
8
5040 15
|22 9960 28
3000
Dy
14400
8
12 5040
22 24 9960
3000
9
= 17280
6
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12 15 |22 24 28 9 The determinant of coefficient matrix, D = 8 6 9(24 x 8 - 28 x 8) - 12(22 x 8 - 28 x 6) 15(22 x 8 - 28 x 6) = 96 Similarly 15 |5040 12 24 28 9960 5760 Dr 3000 8 5040 15 |22 9960 28 3000 Dy 14400 8 12 5040 22 24 9960 3000 9 = 17280 6

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