Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

Building costs A housing contractor has orders for 4 onebedroom units, 10 two- bedroom units, and 6 three-bedroom units. The labor and material costs (in thousands of dollars) are given in the following table. $$ \begin{array}{|l|ccc|} \hline & \text { 1-Bedroom } & \text { 2-Bedroom } & \text { 3-Bedroom } \\ \hline \text { Labor } & 70 & 95 & 117 \\ \text { Materials } & 90 & 105 & 223 \\ \hline \end{array} $$ (a) Organize these data into an order matrix \(A\) and a cost matrix \(B\) so that the product \(C=A B\) is defined. (b) Find \(C\). (c) Interpret the meaning of each element in \(C\).

Step-by-Step Solution

Answer

Total labor cost: $1,932,000, total material cost: $2,748,000.

Step 1: Understand the Problem Requirements

We need to calculate the cost for 4 one-bedroom units, 10 two-bedroom units, and 6 three-bedroom units. We organize the orders into an order matrix \(A\) and the costs into a cost matrix \(B\). The product \(C = AB\) will then give us the total cost for each category.

Step 2: Define Matrix A and Matrix B

Matrix \(A\) represents the number of units of each type, and matrix \(B\) contains the labor and material costs. Define matrices as:\[A = \begin{bmatrix}4 \10 \6\end{bmatrix}, B = \begin{bmatrix}70 & 90 \95 & 105 \117 & 223\end{bmatrix}\]

Step 3: Verify Matrix Multiplication Compatibility

Ensure that the matrices can be multiplied. Matrix \(A\) is a 3x1 matrix and \(B\) is a 3x2 matrix. Since the number of columns in \(A\) (1) matches the number of rows in \(B\) (3), the product \(AB\) is possible and results in a 3x2 matrix.

Step 4: Calculate the Product C = AB

Perform the matrix multiplication:\[C = A \times B = \begin{bmatrix} 4 \times 70 + 10 \times 95 + 6 \times 117 & 4 \times 90 + 10 \times 105 + 6 \times 223 \ = 280 + 950 + 702 & 360 + 1050 + 1338 \ = 1932 & 2748\end{bmatrix} \] Thus, \[C = \begin{bmatrix} 1932 & 2748 \end{bmatrix}\]

Step 5: Interpret Each Element in C

Each element in \(C\) represents the total cost for the ordered units. The first element (1932) is the total labor cost, and the second element (2748) is the total material cost, both calculated in thousands of dollars. Therefore, the total cost of building is \\(1,932,000 for labor and \\)2,748,000 for materials.

Key Concepts

Order Matrix

An order matrix is a mathematical representation used to organize numerical data conveniently. In the context of this exercise, the order matrix \( A \) represents the number of housing units that the contractor needs to build. It is structured as a vertical single-column matrix, indicating the different types of unit orders. Each entry in the matrix corresponds to a specific category of housing:

• First row: 4, representing four one-bedroom units
• Second row: 10, representing ten two-bedroom units
• Third row: 6, representing six three-bedroom units

The order matrix is crucial because it simplifies the organization and calculation of total costs by aligning it with the cost matrix for effective matrix multiplication.

Cost Matrix

The cost matrix \( B \) encapsulates the various costs, both labor and materials, associated with constructing each type of housing unit. It's depicted as a two-column matrix where each column shows a different aspect of the cost, and each row corresponds to a type of unit.

• First column (Labor Cost): Shows labor costs for 1-bedroom, 2-bedroom, and 3-bedroom units as 70, 95, and 117 respectively.
• Second column (Material Cost): Displays material costs as 90, 105, and 223 for each type of unit in order.

This layout efficiently aligns the cost data, allowing it to be easily combined with the order matrix during matrix multiplication to determine the total costs. Each element in matrix \( B \) provides vital information about specific cost parameters needed to compute the grand totals accurately.

Matrix Product Interpretation

Interpreting a matrix product helps in deciphering the results from the multiplication of matrices, a vital skill in real-world applications like budgeting in construction projects. When multiplying the order matrix \( A \) with the cost matrix \( B \), the resultant matrix \( C \) reveals comprehensive cost insights.

• The first element (1932 in matrix \( C \)) represents the complete labor cost of constructing all ordered units, summed up through the individual labor costs across all unit types, expressed in thousands of dollars.
• The second element (2748 in matrix \( C \)) indicates the total material cost. It aggregates the material costs for all housing units together, similarly expressed in thousands of dollars.

Thus, each element in \( C \) communicates a clear financial picture: The total labor and material costs required to fulfill the construction orders of the housing units. This method of calculation not only provides precision in budgeting but also makes it straightforward to adjust and understand cost impact with changes in unit orders or costs.