Page 2 of 6 2 36 points) Apartments \'R Us is a modular construction company tha
ID: 3185433 • Letter: P
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Page 2 of 6 2 36 points) Apartments 'R Us is a modular construction company that has bee design and build a number of high rise apartment buildings in cities across the United States. The company uses the following loor plans, where a single square indicates a 1-bedroom apartment, a Ix2 rectangle indicates a 2-bedroom apartment and a 1x3 rectangle or a 3 square L. shape indicates a 3-bedroom apartment. The number of bedrooms of each type is listed in parentheses (I bedrooms, # 2 bedrooms, # 3 bedrooms). Hallway Hallway lod Hallway C) Elevanor Floor Plan A 3.5. 1) or Plan B (6.2,2) Floor Plan C (1.3,3) a. You want to construct a building that has 2 floors of Plan A, 3 floors of Plan B and 4 floors of Plan C. How many 1-bedroom, 2-bedroom and 3-bedroom apartments will there be in the building? Plan Asp b. If you want to create a building with exactly 17 1-bedroom apartments, 23 2-bedroom apartments and 11 three-bedroom apartments. How many floors of each plan will you need? 7, 23, 11) 3517 62 2 23 paMoue ane suouenope Aod sooedsExplanation / Answer
a. In floor plan A, 3 one bedroom, 5 two bedroom and 1 three bedroom apartments are to be built. This can be represented as a triplet (3,5,1). Similarly, in floor plan B, 6 one bedroom, 2 two bedroom and 2 three bedroom apartments are to be built. This can be represented as a triplet (6,2,2).Also, in floor plan C, 1 one bedroom, 3 two bedroom and 3 three bedroom apartments are to be built. This can be represented as a triplet (1,3,3).
If a building has 2 floors of plan A, 3 floors of plan B and 4 floors of plan C, then the no. of one bedroom, two bedroom and three bedroom apartments is 2(3,5,1)+3(6,2,2)+4(1,3,3) = (6,10,2)+(18,6,6)+(4,12,12) = (28,28,20). Thus, there will be 28 one bedroom, 28 two bedroom and 20 three bedroom apartments in the building.
b. Let there be x floors of plan A, y floors of plan B and z floors of plan C in the building so as to have 17 one bedroom, 23 two bedroom and 11 three bedroom apartments in the building. Then x(3,5,1)+y(6,2,2)+z(1,3,3) =(17,23,11) or, 3x+6y+z = 17, 5x+2y+3z = 23 and x+2y+3z = 11. The augmented matrix of this linear system is A =
3
6
1
17
5
2
3
23
1
2
3
11
To solve the above linear system, we have to reduce A to its RREF which is
1
0
0
3
0
1
0
1
0
0
1
2
This implies that x = 3,y =1 and z = 2. Thus, there will be 3 floors of plan A, 1 floors of plan B and 2 floors of plan C in the building.
3
6
1
17
5
2
3
23
1
2
3
11
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