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Chapter 1: Q28E (page 1)

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.

Short Answer

Expert verified
  1. The amount of heat generated when the steam plant burns\({x_1}\)tons of anthracite and \({x_2}\) tons of bituminous coal is \(27.6{x_1} + 30.2{x_2}\) million Btu.
  2. The total output generated by \({x_1}\) tons of anthracite and \({x_2}\) tons of bituminous coal is represented by the linear combination as \({x_1}\left[ {\begin{array}{*{20}{c}}{27.6}\\{3100}\\{250}\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}{30.2}\\{6400}\\{360}\end{array}} \right]\)
  3. The steam plant consumed 3.9 tonnes of anthracite coal and 1.8 tonnes of bituminous coal.

Step by step solution

01

Determine the amount of heat produced when the steam plant burns

a.

The amount of heat generated when the steam plant burns \({x_1}\) tons of anthracite and \({x_2}\) tons of bituminous coal is \(27.6{x_1} + 30.2{x_2}\) million Btu.

02

Express the total output in a linear combination

b.

A vector equation\({{\mathop{\rm x}\nolimits} _1}{a_1} + {x_2}{a_2} + ... + {x_n}{a_n} = b\)has the same solution set. The vector\({\mathop{\rm y}\nolimits} \)defined by\(y = {c_1}{v_1} + .... + {c_p}{v_p}\)is called alinear combination of\({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_p}\)with weights\({c_1},{c_2},...,{c_p}\).

Assume that the steam plant burns\({x_1}\)tons of anthracite and \({x_2}\) tons of bituminous coal.

The total output generated by\({x_1}\)tons of anthracite and \({x_2}\) tons of bituminous coal is represented by the linear combination as:

\({x_1}\left[ {\begin{array}{*{20}{c}}{27.6}\\{3100}\\{250}\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}{30.2}\\{6400}\\{360}\end{array}} \right]\)

03

Write the linear combination of the vector into an augmented matrix

c.

It is given that the steam plant produced 162 million Btu of heat, 23,610 g of sulphur dioxide, and 1623 g of particulate matter.

The linear combination of the heat of steam plant produced is:

\({x_1}\left[ {\begin{array}{*{20}{c}}{27.6}\\{3100}\\{250}\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}{30.2}\\{6400}\\{360}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{162}\\{23610}\\{1623}\end{array}} \right]\)

The augmented matrix for the obtained equation is represented as:

\(\left[ {\begin{array}{*{20}{c}}{27.6}&{30.2}&{162}\\{3100}&{6400}&{23610}\\{250}&{360}&{1623}\end{array}} \right]\)

04

Apply row operation

Perform an elementary row operation to produce the first augmented matrix.

Multiply row 1 by \(\frac{1}{{27.6}}\).

\(\left[ {\begin{array}{*{20}{c}}1&{1.0942}&{5.869}\\{3100}&{6400}&{23610}\\{250}&{360}&{1623}\end{array}} \right]\)

05

Apply row operation

Perform an elementary row operationto produce the second augmented matrix.

Apply the sum of \( - 3100\) times row 1 and row 2 at row 2, and the sum of \( - 250\) times row 1 and row 3 at row 3.

\(\left[ {\begin{array}{*{20}{c}}1&{1.0942}&{5.869}\\0&{3007.98}&{5416.1}\\0&{86.449}&{155.60}\end{array}} \right]\)

Multiply row 2 by \(\frac{1}{{3007.98}}\).

\(\left[ {\begin{array}{*{20}{c}}1&{1.0942}&{5.869}\\0&1&{1.8}\\0&{86.449}&{155.60}\end{array}} \right]\)

06

Apply row operation 

Perform an elementaryrow operation to produce the third augmented matrix.

Apply the sum of \( - 1.0942\) times row 2 and row 1 at row 1.

\(\left[ {\begin{array}{*{20}{c}}1&0&{3.9}\\0&1&{1.8}\\0&{86.449}&{155.60}\end{array}} \right]\)

Apply the sum of \(86.449\) times row 2 and row 3 at row 3.

\(\left[ {\begin{array}{*{20}{c}}1&0&{3.9}\\0&1&{1.8}\\0&0&0\end{array}} \right]\)

07

Convert the matrix into the equation

If vector\({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_p}\)in\({\mathbb{R}^n}\)are given with scalars\({c_1},{c_2},...,{c_p}\), then the vector\({\mathop{\rm y}\nolimits} \)defined by\(y = {c_1}{v_1} + .... + {c_p}{v_p}\)is called alinear combination of\({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_p}\)with weights\({c_1},{c_2},...,{c_p}\).

To obtain the solution of the vector equations, it is required to convert the augmented matrix into the system of equations.

Write the obtained matrix \(\left[ {\begin{array}{*{20}{c}}1&0&{3.9}\\0&1&{1.8}\\0&0&0\end{array}} \right]\)into the equation notation.

\(\begin{array}{l}{x_1} = 3.9\\{x_2} = 1.8\end{array}\)

Thus, the steam plant consumes 3.9 tonnes of anthracite coal and 1.8 tonnes of bituminous coal.

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Most popular questions from this chapter

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.

In Exercises 5 and 6, follow the method of Examples 1 and 2 to write the solution set of the given homogeneous system in parametric vector form.

5. \(\begin{aligned}{c}{x_1} + 3{x_2} + {x_3} = 0\\ - 4{x_1} - 9{x_2} + 2{x_3} = 0\\ - 3{x_2} - 6{x_3} = 0\end{aligned}\)

Determine by inspection whether the vectors in Exercises 15-20 are linearly independent. Justify each answer.

16. \(\left[ {\begin{array}{*{20}{c}}4\\{ - 2}\\6\end{array}} \right],\left[ {\begin{array}{*{20}{c}}6\\{ - 3}\\9\end{array}} \right]\)

Find an equation involving \(g,\,h,\)and \(k\) that makes this augmented matrix correspond to a consistent system:

\(\left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&g\\0&3&{ - 5}&h\\{ - 2}&5&{ - 9}&k\end{array}} \right]\)

Suppose Tand Ssatisfy the invertibility equations (1) and (2), where T is a linear transformation. Show directly that Sis a linear transformation. (Hint: Given u, v in \({\mathbb{R}^n}\), let \({\mathop{\rm x}\nolimits} = S\left( {\mathop{\rm u}\nolimits} \right),{\mathop{\rm y}\nolimits} = S\left( {\mathop{\rm v}\nolimits} \right)\). Then \(T\left( {\mathop{\rm x}\nolimits} \right) = {\mathop{\rm u}\nolimits} \), \(T\left( {\mathop{\rm y}\nolimits} \right) = {\mathop{\rm v}\nolimits} \). Why? Apply Sto both sides of the equation \(T\left( {\mathop{\rm x}\nolimits} \right) + T\left( {\mathop{\rm y}\nolimits} \right) = T\left( {{\mathop{\rm x}\nolimits} + y} \right)\). Also, consider \(T\left( {cx} \right) = cT\left( x \right)\).)
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