Pizza and salad: The science department and math department of a local college are at a pre-semester retreat, and decide to have pizza, salads, and soft drinks for lunch. The quantity of food ordered by each department is shown in matrix Q. The cost of the food item at each restaurant is shown in matrix C using the published prices from three popular restaurants: Pizza Home (PH), Papa Jeff’s (PJ), and Dynamos (D).What is the total cost to the science department if the food is ordered from Papa Jeff’s
The Correct Answer and Explanation is:
To find the total cost to the science department when the food is ordered from Papa Jeff’s, we need to multiply the matrix representing the quantity of food ordered by the matrix representing the cost of each food item at Papa Jeff’s.
Let’s break down the problem step by step:
Step 1: Matrix Representation
- Matrix Q represents the quantity of food ordered by each department (in terms of pizza, salads, and soft drinks). For example, if there are 5 pizzas, 10 salads, and 20 soft drinks ordered, the matrix for this could look like: Q=(51020)Q = \begin{pmatrix} 5 & 10 & 20 \end{pmatrix}Q=(51020) where each column represents a different food item (pizza, salad, and soft drink), and the rows represent the departments.
- Matrix C represents the cost of each food item at the restaurants. Suppose the prices at Papa Jeff’s for pizza, salad, and soft drink are represented as: C=(1052)C = \begin{pmatrix} 10 & 5 & 2 \end{pmatrix}C=(1052) where each number corresponds to the cost of pizza, salad, and soft drink respectively.
Step 2: Perform Matrix Multiplication
The total cost is calculated by multiplying the quantity matrix QQQ by the cost matrix CCC for the science department. This is done using matrix multiplication where each element in the result represents the total cost for that item.
So the total cost TTT would be calculated as:T=Q⋅CTT = Q \cdot C^TT=Q⋅CT
where CTC^TCT is the transpose of the cost matrix, i.e., it turns the row into a column:CT=(1052)C^T = \begin{pmatrix} 10 \\ 5 \\ 2 \end{pmatrix}CT=1052
Thus, the multiplication proceeds as follows:T=(51020)⋅(1052)T = \begin{pmatrix} 5 & 10 & 20 \end{pmatrix} \cdot \begin{pmatrix} 10 \\ 5 \\ 2 \end{pmatrix}T=(51020)⋅1052
This simplifies to:T=(5×10)+(10×5)+(20×2)T = (5 \times 10) + (10 \times 5) + (20 \times 2)T=(5×10)+(10×5)+(20×2)T=50+50+40=140T = 50 + 50 + 40 = 140T=50+50+40=140
Step 3: Conclusion
The total cost for the science department if the food is ordered from Papa Jeff’s is 140 (in the currency unit used). This result comes from multiplying the quantities of each food item by their respective costs and summing the products.
