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Problem 462

# Suppose a building contractor has accepted orders for 5 ranch style houses, 7 Cape Cod houses and 12 colonial style houses. The table below gives the amount of each raw material going into each type of house expressed in convenient units: 1) Represent the builder's order by a row vector. Represent the table above in matrix form. Find how much steel, wood, glass, paint and labor the building needs. 2) Suppose that steel costs $$\ 15$$ per unit, wood cost $$\ 8$$ per unit, glass costs $$\ 5$$ per unit, paint costs $$\ 1$$ per unit and labor costs $$\ 10$$ per unit. Represent this as a column vector and find the total cost for the builder.

Expert verified
In this exercise, we represent the builder's order as a row vector, $$\textbf{O} =\begin{pmatrix} 5 & 7 & 12 \end{pmatrix}$$, and the raw material requirements for each house type as a matrix, $$\textbf{M}$$. We multiply the order vector with the matrix to find the total amount of each raw material needed, $$\textbf{O}\textbf{M}$$. Then, we represent the cost of each raw material as a column vector, $$\textbf{C}$$, and multiply it with the total raw material requirements to find the total cost for the builder, $$(\textbf{O}\textbf{M})\textbf{C}$$.
See the step by step solution

## Step 1: Represent the builder's order as a row vector

The building contractor has accepted orders for 5 ranch style houses, 7 Cape Cod houses, and 12 colonial style houses. We can represent this order as a row vector: $\textbf{O} = \begin{pmatrix} 5 & 7 & 12 \end{pmatrix}$

## Step 2: Represent the table as a matrix

Now, we represent the table of raw material requirements for each house type as a matrix, where the rows correspond to materials and the columns correspond to the house types: $\textbf{M} = \begin{pmatrix} \text{steel} \\ \text{wood} \\ \text{glass} \\ \text{paint} \\ \text{labor} \end{pmatrix} \begin{pmatrix} \text{ranch} \\ \text{Cape Cod} \\ \text{colonial} \end{pmatrix} = \begin{pmatrix} a & d & g \\ b & e & h \\ c & f & i \\ j & k & l \\ m & n & o \end{pmatrix}$ Replace the variables (a, b, c, ...) with the actual amounts of raw materials required for each house type from the table given in the exercise.

## Step 3: Multiply the builder's order vector with the matrix

Now, we multiply the builder's order vector with the raw material matrix to find the total raw materials required: $\textbf{O}\textbf{M} = \begin{pmatrix} 5 & 7 & 12 \end{pmatrix}\begin{pmatrix} a & d & g \\ b & e & h \\ c & f & i \\ j & k & l \\ m & n & o \end{pmatrix}$ Calculate the result of the matrix multiplication to obtain the total amount of each raw material needed.

## Step 4: Represent costs as a column vector

Now we represent costs of steel, wood, glass, paint, and labor as a column vector: $\textbf{C} = \begin{pmatrix} \15 \\ \8 \\ \5 \\ \1 \\ \10 \end{pmatrix}$

## Step 5: Find the total cost for the builder

Finally, we multiply the total raw materials vector obtained in Step 3 with the cost vector: $\text{Total cost} = (\textbf{O}\textbf{M})\textbf{C}$ Calculate the result of the matrix multiplication to find the total cost for the builder.

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