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Chapter 1: Q25Q (page 1)

A large apartment building is to be built using modular construction techiniques. The arrangement of apartment on any particular floor is to be chosen from one of three basic floor plans. Plan A has 18 apartments on one floor, including 3 three bedroom units, and 8 one bedroom units. 7 two bedroom units and 8 one bedroom units. Each floor of plan B includes 4 three bedroom units, 4 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 \({x_{\bf{1}}}\) floors of plan A, \({x_2}\) floor of plans B, and \({x_{\bf{3}}}\) floors of plan C.

a. What interpretation can be given to the vector \({x_{\bf{1}}}\left( {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{7}}\\{\bf{8}}\end{aligned}} \right)\)?

b. Write a formal linear combination of vectors that expresses the total numbers of three-, two-, and one-bedroom apartments contained in the building.

c. (M) Is it possible to design the building with exactly 66 three bedroom units, 74 two bedrooms units, and 136 one bedroom units? If so, is there more than one way to do it? Explain your answer.

Short Answer

Expert verified

a. Vector \({x_1}\) represents the number of bedrooms.

b. \({x_1}\left( {\begin{aligned}{*{20}{c}}3\\7\\8\end{aligned}} \right) + {x_2}\left( {\begin{aligned}{*{20}{c}}4\\4\\8\end{aligned}} \right) + {x_3}\left( {\begin{aligned}{*{20}{c}}5\\3\\9\end{aligned}} \right)\)

c. There are only two possible plans:

(i) Two floors of plan A, 15 floors of plan B, and zero floors of plan C

(ii) Six floors of plan A, two floors of plan B, and eight floors of plan C

The other solution for the equation of \({x_1}\), \({x_2}\) and, \({x_3}\) has negative results, which is not possible for the number of floors.

Step by step solution

01

Interpret \({x_{\bf{1}}}\)

The vector \({x_1}\left( {\begin{aligned}{*{20}{c}}3\\7\\8\end{aligned}} \right)\) represents the number of three-, two-, and one- bedroom apartments on floor \({x_1}\).

02

Find the linear combination of vectors to express the total number of apartments

For floors \({x_1}\), \({x_2}\), and \({x_3}\), the linear combination of vectors is

\({x_1}\left( {\begin{aligned}{*{20}{c}}3\\7\\8\end{aligned}} \right) + {x_2}\left( {\begin{aligned}{*{20}{c}}4\\4\\8\end{aligned}} \right) + {x_3}\left( {\begin{aligned}{*{20}{c}}5\\3\\9\end{aligned}} \right)\).

03

Form the equation for the total number of apartments in the building

By the linear combination of vectors, you get

\({x_1}\left( {\begin{aligned}{*{20}{c}}3\\7\\8\end{aligned}} \right) + {x_2}\left( {\begin{aligned}{*{20}{c}}4\\4\\8\end{aligned}} \right) + {x_3}\left( {\begin{aligned}{*{20}{c}}5\\3\\9\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}{66}\\{74}\\{136}\end{aligned}} \right)\).

04

Form the augmented matrix

Using the equation for the linear combination of vectors, the augmented matrix is

\(\left( {\begin{aligned}{*{20}{c}}3&4&5&{66}\\7&4&3&{74}\\8&8&9&{136}\end{aligned}} \right)\).

05

Convert the matrix into the echelon form

Use the code in the MATLAB to obtain the row-reduced echelon form of the augmented matrix \(\left( {\begin{aligned}{*{20}{c}}3&4&5&{66}\\7&4&3&{74}\\8&8&9&{136}\end{aligned}} \right)\).

\(\begin{aligned}{l} > > {\rm{ A }} = {\rm{ }}\left( {{\rm{3 4 5 66}};{\rm{ 7 4 3 74}};{\rm{ 8 8 9 136}};} \right);\\ > > {\rm{ U}} = {\rm{rref}}\left( {\rm{A}} \right)\end{aligned}\)

\(\left( {\begin{aligned}{*{20}{c}}1&0&{ - \frac{1}{2}}&{ - 2}\\0&1&{\frac{{13}}{8}}&{ - 15}\\0&0&0&0\end{aligned}} \right)\)

Then, \({x_1} - \frac{1}{2}{x_3} = 2\) and \({x_2} + \frac{{13}}{8}{x_3} = 15\).

06

Find the general solution

If \({x_3} = 0\), then there are two floors of plan A and 15 floors of plan B.

If \({x_3} = 8\), then there are six floors of plan A, two floors of plan B, and eight floors of plan C.

These are the only feasible solutions. For larger multiples of 8, the number of floors will be negative.

Hence, \({x_1}\) represents the number of different bedrooms. The linear combination of floor vectors is \({x_1}\left( {\begin{aligned}{*{20}{c}}3\\7\\8\end{aligned}} \right) + {x_2}\left( {\begin{aligned}{*{20}{c}}4\\4\\8\end{aligned}} \right) + {x_3}\left( {\begin{aligned}{*{20}{c}}5\\3\\9\end{aligned}} \right)\), and the possible number of floors are:

  1. Two floors of plan A, 15 floors of plan B, and zero floors of plan C
  2. Six floors of plan A, two floors of plan B, and eight floors of plan C

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Most popular questions from this chapter

In Exercises 21 and 22, mark each statement True or False. Justify each answer on the basis of a careful reading of the text.

21.

a. The columns of a matrix \(A\) are linearly independent if the equation \(Ax = 0\) has a trivial solution.

b. If \(S\) is a linearly dependent set, then each vector is a linear combination of the other vectors in \(S\).

c. The columns of any \(4 \times 5\) matrix are linearly dependent.

d. If \({\mathop{\rm x}\nolimits} \) and \(y\) are linearly independent, and if \(\left\{ {x,y,z} \right\}\) is linearly dependent, then \(z\) is in Span\(\left\{ {x,y} \right\}\).

A steam plant burns two types of coal: anthracite (A) and bituminous (B). For each ton of A burned, the plant produces 27.6 million of Btu of heat, 3100 grams (g) of sulphur dioxide, and 250g of particulate matter (solid-particle pollutants). For each ton of B burned, the plant produces 30.2 million Btu, 6400g of sulphur dioxide, and 360g of particulate matter.

  1. How much heat does the steam plant produce when it burns \({x_1}\) tons of \(A\) and \({x_2}\) tons of \(B\).
  2. Suppose a vector that lists the amounts of heat, sulphur dioxide, and particulate matter describes the output of the steam plant. Express this output as a linear combination of two vectors, assuming that the plant burns \({x_1}\) tons of \(A\) and \({x_2}\) tons of \(B\).
  3. [M] Over a certain time period, the steam plant produced 162 million Btu of heat, 23,610 g of sulphur dioxide, and 1623 g of particulate matter. Determine how many tons of each type of coal the steam plant must have burned. Include a vector equation as part of your solution.

In Exercises 23-26, describe the possible echelon forms of the matrix. Use the notation of Example 1 in Section 1.2

23. \(A\) is a \(3 \times 3\) matrix with linearly independent columns.

In Exercises 17-20, show that \(T\) is a linear transformation by finding a matrix that implements the mapping. Note that \({x_1}\), \({x_2}\),鈥︹�︹�� are not vectors but are enteries in vectors

\(T\left( {{x_1},{x_2}} \right) = \left( {2{x_2} - 3{x_1},{x_1} - 4{x_2},0,{x_2}} \right)\)

In a certain region, about 6% of city鈥檚 population moves to the surrounding suburbs each year, and about 4% of the suburban population moves into the city. In 2015, there were, 10,000,000 residents in the city and 800,000 in the suburbs. Set up a difference equation that describes this situation, where\({{\bf{x}}_{\bf{0}}}\)is the initial population in 2015. Then estimate the population in the city and in the suburbs two years later, in 2017.
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