5) A building in an apartment complex is to be built using a new modular technique. The arrangement of apartments on any floor will be chosen from one of three floor plans. The floorplan for Plan A has 4 three- bedroom units, 4 two-bedroom units, and 8 one-bedroom units. Each floor of plan B includes 3 three-bedroom units, 7 two-bedroom units, and 8 one-bedroom units. Each floor of plan C includes 5 three-bedroom units, 3 two-bedroom units, and 9 one-bedroom units. Suppose the building contains a total of x, floors of plan A, x2 floors of plan B, and x3 floors of plan C. 4 a) What does the vector xX1| 4 represent? 8 b) Write a linear combination of vectors that expresses the total numbers of three-, two-, and one-bedroom apartments contained in the building. c) Is it possible to design the building with exactly 66 three-bedroom units, 74 two-bedroom units, and 136 one- bedroom units? If so, is there more than one way to do it? Explain your answer.
5) A building in an apartment complex is to be built using a new modular technique. The arrangement of apartments on any floor will be chosen from one of three floor plans. The floorplan for Plan A has 4 three- bedroom units, 4 two-bedroom units, and 8 one-bedroom units. Each floor of plan B includes 3 three-bedroom units, 7 two-bedroom units, and 8 one-bedroom units. Each floor of plan C includes 5 three-bedroom units, 3 two-bedroom units, and 9 one-bedroom units. Suppose the building contains a total of x, floors of plan A, x2 floors of plan B, and x3 floors of plan C. 4 a) What does the vector xX1| 4 represent? 8 b) Write a linear combination of vectors that expresses the total numbers of three-, two-, and one-bedroom apartments contained in the building. c) Is it possible to design the building with exactly 66 three-bedroom units, 74 two-bedroom units, and 136 one- bedroom units? If so, is there more than one way to do it? Explain your answer.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Text Transcription for Educational Website:**
**5) Modular Apartment Building Design**
A building in an apartment complex is to be constructed using a new modular technique. The arrangement of apartments on any floor will be chosen from one of three floor plans.
- **Floorplan for Plan A** has:
- 4 three-bedroom units
- 4 two-bedroom units
- 8 one-bedroom units
- **Floorplan for Plan B** includes:
- 3 three-bedroom units
- 7 two-bedroom units
- 8 one-bedroom units
- **Floorplan for Plan C** includes:
- 5 three-bedroom units
- 3 two-bedroom units
- 9 one-bedroom units
Suppose the building contains a total of \(x_1\) floors of plan A, \(x_2\) floors of plan B, and \(x_3\) floors of plan C.
**a) Analyze the Vector**
What does the vector \(\begin{bmatrix} 4 \\ 4 \\ 8 \end{bmatrix}\) represent?
- **Explanation**: This vector represents the number of three-bedroom, two-bedroom, and one-bedroom units in Plan A, respectively.
**b) Linear Combination of Vectors**
Write a linear combination of vectors that expresses the total numbers of three-, two-, and one-bedroom apartments contained in the building.
- **Solution**: The total number of apartment types can be expressed as a vector equation:
\[
\begin{bmatrix} \text{Total 3-bedroom units} \\ \text{Total 2-bedroom units} \\ \text{Total 1-bedroom units} \end{bmatrix} = x_1 \begin{bmatrix} 4 \\ 4 \\ 8 \end{bmatrix} + x_2 \begin{bmatrix} 3 \\ 7 \\ 8 \end{bmatrix} + x_3 \begin{bmatrix} 5 \\ 3 \\ 9 \end{bmatrix}
\]
**c) Design Feasibility**
Is it possible to design the building with exactly 66 three-bedroom units, 74 two-bedroom units, and 136 one-bedroom units? If so, is there more than one way to do it? Explain your answer.
- **Solution**: To find if it's possible, solve the following system of equations](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7803273c-2063-4cef-bfbb-565e22329fee%2Ffe51a9b0-f110-489c-ab59-0fffd868bfe0%2F0fhxqz_processed.png&w=3840&q=75)
Transcribed Image Text:**Text Transcription for Educational Website:**
**5) Modular Apartment Building Design**
A building in an apartment complex is to be constructed using a new modular technique. The arrangement of apartments on any floor will be chosen from one of three floor plans.
- **Floorplan for Plan A** has:
- 4 three-bedroom units
- 4 two-bedroom units
- 8 one-bedroom units
- **Floorplan for Plan B** includes:
- 3 three-bedroom units
- 7 two-bedroom units
- 8 one-bedroom units
- **Floorplan for Plan C** includes:
- 5 three-bedroom units
- 3 two-bedroom units
- 9 one-bedroom units
Suppose the building contains a total of \(x_1\) floors of plan A, \(x_2\) floors of plan B, and \(x_3\) floors of plan C.
**a) Analyze the Vector**
What does the vector \(\begin{bmatrix} 4 \\ 4 \\ 8 \end{bmatrix}\) represent?
- **Explanation**: This vector represents the number of three-bedroom, two-bedroom, and one-bedroom units in Plan A, respectively.
**b) Linear Combination of Vectors**
Write a linear combination of vectors that expresses the total numbers of three-, two-, and one-bedroom apartments contained in the building.
- **Solution**: The total number of apartment types can be expressed as a vector equation:
\[
\begin{bmatrix} \text{Total 3-bedroom units} \\ \text{Total 2-bedroom units} \\ \text{Total 1-bedroom units} \end{bmatrix} = x_1 \begin{bmatrix} 4 \\ 4 \\ 8 \end{bmatrix} + x_2 \begin{bmatrix} 3 \\ 7 \\ 8 \end{bmatrix} + x_3 \begin{bmatrix} 5 \\ 3 \\ 9 \end{bmatrix}
\]
**c) Design Feasibility**
Is it possible to design the building with exactly 66 three-bedroom units, 74 two-bedroom units, and 136 one-bedroom units? If so, is there more than one way to do it? Explain your answer.
- **Solution**: To find if it's possible, solve the following system of equations
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