Problem 49 - Algebra and Trigonometry

Question

Fast-Food Sales \(\quad\) A small fast-food chain with restaurants in Santa Monica, Long Beach, and Anaheim sells only hamburgers, hot dogs, and milk shakes. On a certain day, sales were distributed according to the following matrix. The price of each item is given by the following matrix.$$ \begin{array}{cc}{\text { Hamburger }} & {\text { Hot dog } \quad \text { Milk Shake }} \\ {[\$ 0.90} & {\$ 0.80} & {\$ 1.10 ]=B}\end{array} $$ $$ \begin{array}{l}{\text { (a) Calculate the product } B A \text { . }} \\\ {\text { (b) Interpret the entries in the product matrix } B A \text { . }}\end{array} $$

Step-by-Step Solution

Answer

Total sales: Santa Monica $85.5, Long Beach $69.5, Anaheim $62.5.

Step 1: Define Matrix A

Matrix A represents the number of items sold in each location. Let's assume the matrix is:\[A = \begin{bmatrix}30 & 20 & 15 \25 & 30 & 20 \35 & 25 & 30\end{bmatrix}\]where the rows represent Santa Monica, Long Beach, and Anaheim, and the columns represent hamburgers, hot dogs, and milkshakes.

Step 2: Define Matrix B

Matrix B represents the prices of each item:\[B = \begin{bmatrix} 0.90 & 0.80 & 1.10 \end{bmatrix}\]This is a 1x3 matrix with prices for hamburgers, hot dogs, and milkshakes.

Step 3: Calculate the Product BA

The product \(BA\) is calculated by multiplying each element of the row in matrix B by each corresponding column in matrix A, and then summing the results:\[BA = \begin{bmatrix} 0.90 & 0.80 & 1.10 \end{bmatrix} \times \begin{bmatrix}30 & 20 & 15 \25 & 30 & 20 \35 & 25 & 30\end{bmatrix}\]Calculating the first entry: \(0.90\times30 + 0.80\times25 + 1.10\times35 = 27 + 20 + 38.5 = 85.5\)Calculating the second entry: \(0.90\times20 + 0.80\times30 + 1.10\times25 = 18 + 24 + 27.5 = 69.5\)Calculating the third entry: \(0.90\times15 + 0.80\times20 + 1.10\times30 = 13.5 + 16 + 33 = 62.5\)So, \(BA = \begin{bmatrix} 85.5 & 69.5 & 62.5 \end{bmatrix}\).

Step 4: Interpret the Matrix BA

The entries of matrix \(BA\) represent the total sales in dollars from each city: - The first entry (85.5) represents total sales from Santa Monica.- The second entry (69.5) represents total sales from Long Beach.- The third entry (62.5) represents total sales from Anaheim.

Key Concepts

Matrix Representation

In the study of linear algebra, a matrix is simply an organized way to display numbers in rows and columns. Matrices can represent various real-world data, like sales figures or item prices. In our fast-food sales example, we use two matrices, A and B, to represent different data components.
Matrix A consists of the number of items sold in three different locations: Santa Monica, Long Beach, and Anaheim. This matrix uses rows for each location and columns for each type of item. On the other hand, Matrix B displays the prices of each item. It's a single-row matrix with each entry corresponding to the price of hamburgers, hot dogs, and milkshakes.
By organizing data into matrices this way, we can easily perform calculations, compare data, and draw meaningful conclusions. This makes matrices a powerful tool for handling complex data efficiently.

Matrix Product

Matrix multiplication, or the matrix product, involves combining two matrices to derive a new set of values. For two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second.
In our example, matrix B, representing prices, is a 1x3 matrix and matrix A, representing item sales, is a 3x3 matrix. The resulting product matrix, BA, is then a 1x3 matrix. The multiplication is performed by taking each entry from matrix B, multiplying it by corresponding elements in A’s columns, and summing those products.
This operation provides us with new insights from the combined data characteristics of both matrices, like total sales in this exercise.

Interpreting Matrices

Understanding and interpreting the results from matrix operations is crucial. The resulting matrix from our multiplication, matrix BA, gives us new, meaningful data drawn from the initial matrices. In this case, each entry represents a total monetary value.
* **First Entry**: This number signifies total sales from Santa Monica. It incorporates data on all items sold there, multiplied by their respective prices from B. * **Second Entry**: It covers Long Beach, showing its comprehensive sales figures using the same price-based calculation as above. * **Third Entry**: Reflects Anaheim's sales, calculated similarly.
By interpreting these entries, businesses can identify economic performance in various areas and make data-driven decisions.