2. -/1 pointsHarMathAp11 3.4.057. Set up the system of equations and then solve
ID: 3186137 • Letter: 2
Question
2. -/1 pointsHarMathAp11 3.4.057. Set up the system of equations and then solve it by using an inverse matrix. A manufacturer of table saws has three models, Deluxe, Premium, and Ultimate, which must be painted, assembled, and packaged for shipping. The table gives the number of hours required for each of these operations for each type of table saw Deluxe Premium Ultimate Painting1.6 Assembly 2 Packaging 0.5 2 2.4 4 0.5 (a) If the manufacturer has 96 hours avalilable per day for painting, 152 hours for assembly, and 36 hours for packaging, how many of each type of saw can be produced each day? Deluxe saws saws saws Premium Ultimate (b) If 8 more hours of painting time become available, find the new production strategy. Deluxe Premium Ultimate saws saws sawsExplanation / Answer
2.(a). Let the no. of Deluxe, Premium and Ultimate models that can be produced be x,y and z respectively. Then, 1.6x+2y+2.4z = 96 or,(on dividing both the sides by 0.4) 4x +5y+6z= 240…(1), 2x+3y+4z = 152…(2) and 0.5x+0.5y +z = 36 or,( on mutiplying both the sides by 2) x+y+2z = 72…(3).
The augmented matrix of this linear system is A =
4
5
6
240
2
3
4
152
1
1
2
72
To solve the above linear system, we will reduce A to its RREF as under:
Multiply the 1st row by ¼
Add -2 times the 1st row to the 2nd row
Add -1 times the 1st row to the 3rd row
Multiply the 2nd row by 2
Add 1/4 times the 2nd row to the 3rd row
Add -2 times the 3rd row to the 2nd row
Add -3/2 times the 3rd row to the 1st row
Add -5/4 times the 2nd row to the 1st row
Then the RREF of A is
1
0
0
8
0
1
0
8
0
0
1
28
Thus, the solution to the above linear system is x = 8,y = 8,and z = 28. Hence, the no. of Deluxe, Premium and Ultimate models that can be produced are 8,8 and 28 respectively.
(b). If 8 more hours of paining are available, then the 1st equation changes to 1.6x+2y+2.4z = 104 or,(on dividing both the sides by 0.4) 4x +5y+6z= 246…(1). The 2nd and 3rd equations remain unchanged.
The augmented matrix of the new linear system is M =
4
5
6
260
2
3
4
152
1
1
2
72
To solve the above linear system, we will reduce M to its RREF which is
1
0
0
28
0
1
0
8
0
0
1
18
Thus, the solution to the new linear system is x = 28,y = 8,and z = 18. Hence, the no. of Deluxe, Premium and Ultimate models that can be produced are 28,8 and 18 respectively.
4
5
6
240
2
3
4
152
1
1
2
72
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