Question

Computing the cost of Production The Acme Steel Company is a producer of stainless steel and aluminum containers. On a certain day, the following stainless steel containers were manufactured: 500 with 10 -gallon (gal) capacity, 350 with 5-gal capacity, and 400 with 1-gal capacity. On the same day, the following aluminum containers were manufactured: 700 with 10-gal capacity, 500 with 5-gal capacity, and 850 with 1-gal capacity. (a) Find a 2 by 3 matrix representing these data. Find a 3 by 2 matrix to represent the same data. (b) If the amount of material used in the 10 -gal containers is 15 pounds (lb), the amount used in the 5-gal containers is 8 lb, and the amount used in the 1-gal containers is 3 lb, find a 3 by 1 matrix representing the amount of material used. (c) Multiply the 2 by 3 matrix found in part (a) and the 3 by 1 matrix found in part (b) to get a 2 by 1 matrix showing the day's usage of material. (d) If stainless steel costs Acme \(\$ 0.10\) per pound and aluminum costs \(\$ 0.05\) per pound, find a 1 by 2 matrix representing cost. (e) Multiply the matrices found in parts (c) and (d) to find the total cost of the day's production.

Short answer

The total cost of the day's production is $2052.50.

Step-by-step solution

Represent production data as a 2 by 3 matrix

Create a matrix for stainless steel and aluminum production data.\[ Matrix_1 = \begin{pmatrix} 500 & 350 & 400 \ 700 & 500 & 850 \end{pmatrix} \] Stainless steel: 500 (10-gal), 350 (5-gal), 400 (1-gal) Aluminum: 700 (10-gal), 500 (5-gal), 850 (1-gal)

Represent production data as a 3 by 2 matrix

Transpose the previously created matrix. Hence, it will be: \[ Matrix_2 = \begin{pmatrix} 500 & 700 \ 350 & 500 \ 400 & 850 \end{pmatrix} \]

Create a 3 by 1 matrix for the material usage

Given material usage for different containers, the matrix will be: \[ Matrix_3 = \begin{pmatrix} 15 \ 8 \ 3 \end{pmatrix} \]

Multiply 2 by 3 matrix with 3 by 1 matrix

Multiply Matrix_1 and Matrix_3 to find the materials used:\[ Matrix_1 \cdot Matrix_3 = \begin{pmatrix} 500 & 350 & 400 \ 700 & 500 & 850 \end{pmatrix} \cdot \begin{pmatrix} 15 \ 8 \ 3 \end{pmatrix} = \begin{pmatrix} 500 \cdot 15 + 350 \cdot 8 + 400 \cdot 3 \ 700 \cdot 15 + 500 \cdot 8 + 850 \cdot 3 \end{pmatrix} = \begin{pmatrix} 11750 \ 17550 \end{pmatrix} \] Resulting in: \[ \begin{pmatrix} 11750 \ 17550 \end{pmatrix} \]

Create a 1 by 2 matrix for costs

Represent the cost per pound for stainless steel and aluminum: \[ Matrix_4 = \begin{pmatrix} 0.10 & 0.05 \end{pmatrix} \]

Multiply the resulting material usage with cost matrix

Multiply the resulting 2 by 1 matrix from Step 4 with the 1 by 2 cost matrix: \[ Matrix_5 = Matrix_4 \cdot \begin{pmatrix} 11750 \ 17550 \end{pmatrix} = \begin{pmatrix} 0.10 \ 0.05 \end{pmatrix} \cdot \begin{pmatrix} 11750 \ 17550 \end{pmatrix} = 11750 \cdot 0.10 + 17550 \cdot 0.05 = 1175 + 877.5 = 2052.5\]

Key concepts

Matrix Representation

When dealing with data such as production numbers, representing it through matrices can simplify many calculations. In this problem, data for the production of containers by Acme Steel Company is presented. We have two types of containers: stainless steel and aluminum, each with different capacities (10-gallon, 5-gallon, and 1-gallon). To represent this data as a matrix, we create a 2 by 3 matrix where each row stands for a material type and each column for a capacity.
Here's how we organize it:
- The first row contains the numbers for stainless steel containers: 500 (10-gallon), 350 (5-gallon), and 400 (1-gallon).
- The second row contains the numbers for aluminum containers: 700 (10-gallon), 500 (5-gallon), and 850 (1-gallon).
Thus, our 2 by 3 matrix is:
\[ Matrix_1 = \begin{pmatrix} 500 & 350 & 400 \ 700 & 500 & 850 \end{pmatrix} \] This organization makes it easier to perform subsequent calculations, such as determining material usage and production costs. Additionally, the data can be transposed to form a 3 by 2 matrix if needed, where the rows represent container capacities and columns represent material types.

Production Cost Calculation

Calculating production costs involves more than just tallying the number of containers produced. We must also consider the material used and its cost per unit.
For instance, if stainless steel costs \$0.10\ per pound and aluminum costs \$0.05\ per pound, and we know the total material used in pounds for a day's production, we can then compute the overall cost. First, let's multiply the matrix of production data by the matrix of material usage to determine the total material required.
Here is the calculation:
\[ Matrix_1 \cdot Matrix_3 = \begin{pmatrix} 500 & 350 & 400 \ 700 & 500 & 850 \end{pmatrix} \cdot \begin{pmatrix} 15 \ 8 \ 3 \end{pmatrix} = \begin{pmatrix} 11750 \ 17550 \end{pmatrix} \] The resulting matrix shows 11750 pounds of stainless steel and 17550 pounds of aluminum used.
Next, we calculate the cost by multiplying the material usage matrix by the cost per pound matrix:
\[ Matrix_4 \cdot \begin{pmatrix} 11750 \ 17550 \end{pmatrix} = \begin{pmatrix} 0.10 & 0.05 \end{pmatrix} \cdot \begin{pmatrix} 11750 \ 17550 \end{pmatrix} = 11750 \cdot 0.10 + 17550 \cdot 0.05 = 1175 + 877.5 = 2052.5 \] Therefore, the total cost of the day's production is \$2052.50\.

Material Usage Computation

Let's dive deeper into how we compute the material usage for the containers. Each type of container uses a different amount of material, which we need to account for to get an accurate measure.
- A 10-gallon container uses 15 pounds of material.
- A 5-gallon container uses 8 pounds of material.
- A 1-gallon container uses 3 pounds of material.
To represent this, we use a 3 by 1 matrix:
\[ Matrix_3 = \begin{pmatrix} 15 \ 8 \ 3 \end{pmatrix} \] By multiplying this matrix with our production data matrix, we can then determine the total material used. The process is fairly straightforward, as it involves matrix multiplication where each element in the resulting matrix is the sum of the products of elements from the corresponding rows and columns of the original matrices. For example:
\[ Matrix_1 \cdot Matrix_3 = \begin{pmatrix} 500 & 350 & 400 \ 700 & 500 & 850 \end{pmatrix} \cdot \begin{pmatrix} 15 \ 8 \ 3 \end{pmatrix} = \begin{pmatrix} 11750 \ 17550 \end{pmatrix} \] This multiplication breaks down as follows:
- Stainless steel: \ 500 \cdot 15 + 350 \cdot 8 + 400 \cdot 3 = 11750 \ pounds
- Aluminum: \ 700 \cdot 15 + 500 \cdot 8 + 850 \cdot 3 = 17550 \ pounds
This result helps in knowing exactly how much material has been used, a critical step before moving on to cost calculations.