Explanation Given: A manufacturer of swing sets has three model, Deluxe, Premium, and Ultimate, which must be painted, assembled, and packaged for shipping. The following table gives the number of hours required for each of these operations for each type of swing set. Deluxe Premium Ultimate Painting 0.8 1 1.4 Assembly 1 1.5 2 Packaging 0.6 0.75 1 If the manufactures has 55 hours available per day for painting, 75 hours for assembly, and 40 hours for packaging. Definition used: 1. Any augmented matrix that has 1`s or 0`s on the diagonal of its coefficient part and 0`s below the diagonal is said to be in row-echelon form. 2. The augmented matrix representing n equations in n variables is said to be in reduced row-echelon form if it has 1`s or 0`s on the diagonal of its coefficient part and 0`s everywhere else. Calculation: Let the manufactures company produces x units of deluxe swings, y units of premium swings, and z units of ultimate swings. Given that the manufactures has 55 hours available per day for painting, 75 hours for assembly, and 40 hours for packaging. Hence, the detailed summary of the three types of swings and the total hours needed in Assembly, Testing, and Packing listed below in Table 1. Deluxe Premium Ultimate Total hours available Painting 0.8 1 1.4 55 Assembly 1 1.5 2 75 Packaging 0.6 0.75 1 40 Table 1 From Table 1, the equation that represents the total hours needed daily for painting to manufactured the three types of swings is 0.8 x + y + 1.4 z = 55 , since the manufactures has 55 hours available per day for painting. Similarly, the equation that represents the total hours needed daily for assembly to manufactured the three types of swings is x + 1.5 y + 2 z = 75 , since the manufactures has 75 hours available per day for assembly. And the equation that represents the total hours needed daily for packaging to manufactured the three types of swings is 0.6 x + 0.75 y + z = 40 , since the manufactures has 40 hours available per day for packaging. Combine the above equations and form the system of equations in variable x , y , and z as follows. { 0.8 x + y + 1.4 z = 55 x + 1.5 y + 2 z = 75 0.6 x + 0.75 y + z = 40 The augmented matrix associated with the system of linear equations in x , y , and z is [ 0.8 1 1.4 55 1 1.5 2 75 0.6 0.75 1 40 ] . Observe that the augmented matrix is not in row-echelon form. Hence applying certain row operations and reduce it into row-echelon form as follows. Perform the row operation R 2 ↔ R 1 , and obtain the new equivalent matrix. [ 1 1.5 2 75 0.8 1 1.4 55 0.6 0.75 1 40 ] Perform the row operations R 2 − 0.8 R 1 → R 2 and R 3 − 0

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Author: HARSHBARGER

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Chapter 7.2, Problem 38E

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To find: The number of each type swing set produced each day.

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Ch. 7.1 - In Exercises 14, solve the systems of equations....Ch. 7.1 - Prob. 2E Ch. 7.1 - Prob. 3E Ch. 7.1 - Prob. 4E Ch. 7.1 - Prob. 5E Ch. 7.1 - Prob. 6E Ch. 7.1 - Prob. 7E Ch. 7.1 - Prob. 8E Ch. 7.1 - Prob. 9E Ch. 7.1 - Prob. 10E

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The number of each type swing set produced each day.

Solution Summary: The author calculates the hours required for each of the three types of swing sets.

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To find: The number of each type swing set produced each day.

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