Problem 55 - Precalculus: Mathematics for Calculus

Question

Frozen-Food Revenue Some of the frozen foods that Joe's Specialty Foods sells are pesto pizza, spinach ravioli, and macaroni and cheese. The sales distribution for these products is tabulated in matrix \(A\). The retail price (in dollars) for each item is tabulated in matrix \(B\). (a) Calculate the product matrix \(A B\) (b) What is the total revenue for Monday? (c) What is the total revenue from all three days? $$\begin{array}{rccc} & \text { Specialty Food } \\ \hline \text { Pizza } & \text { Ravioli } & \text { Mac and Chesse } \end{array}$$ $$\begin{aligned}&\begin{array}{c}\text { Monday } \\ \text { Tuesday } \\\\\ \text { Wednesday } \end{array}\left[\begin{array}{r} 50 & 20 & 15 \\ 40 & 75 & 20 \\ 35 & 60 & 100 \end{array}\right]=A\end{aligned}$$ $$\begin{aligned}&\qquad\qquad\qquad\quad\text { Price } ($)\\\&\begin{array}{c}\text { Pizza }\\\ \text { Ravioli } \\ \text { Mac and Chesse } \end{array}\left[\begin{array}{r}4 \\\20 \\\10\end{array}\right]=B\end{aligned}$$

Step-by-Step Solution

Answer

The total revenue for Monday is 750 dollars. The total revenue from all three days is 4950 dollars.

Step 1: Understand the Matrices

Matrix \(A\) represents the sales distribution of three frozen food items (pesto pizza, spinach ravioli, macaroni and cheese) over three days: Monday, Tuesday, and Wednesday. Matrix \(B\) contains the retail prices for each item (pizza: \(4, ravioli: \)20, mac and cheese: $10). Our goal is to calculate the revenue generated from sales.

Step 2: Setup for Matrix Multiplication

To find the total revenue for each weekday, we will calculate the matrix product \(AB\). Matrix \(A\) is a \(3 \times 3\) matrix and matrix \(B\) is a \(3 \times 1\) matrix. The resulting product \(AB\) will be a \(3 \times 1\) matrix that represents the total revenue for Monday, Tuesday, and Wednesday respectively.

Step 3: Multiply Matrix \(A\) by Matrix \(B\)

Perform matrix multiplication to find \(AB\). Each element in \(AB\) is calculated by multiplying the corresponding elements of the rows from matrix \(A\) with the column in matrix \(B\) and summing the results.For Monday (1st row of \(A\)): \[(50 \times 4) + (20 \times 20) + (15 \times 10)\] For Tuesday (2nd row of \(A\)): \[(40 \times 4) + (75 \times 20) + (20 \times 10)\] For Wednesday (3rd row of \(A\)): \[(35 \times 4) + (60 \times 20) + (100 \times 10)\]

Step 4: Calculate Numeric Results

Now calculate the numeric results for each day.Monday:\[(50 \times 4) + (20 \times 20) + (15 \times 10) = 200 + 400 + 150 = 750\]Tuesday:\[(40 \times 4) + (75 \times 20) + (20 \times 10) = 160 + 1500 + 200 = 1860\]Wednesday:\[(35 \times 4) + (60 \times 20) + (100 \times 10) = 140 + 1200 + 1000 = 2340\]

Step 5: Total Revenue for Monday

From the calculated matrix product, the total revenue for Monday is 750 dollars.

Step 6: Total Revenue for All Three Days

To find the total revenue from all three days, sum the results from the matrix product \(AB\):\[750 + 1860 + 2340 = 4950\]Thus, the total revenue from Monday to Wednesday is 4950 dollars.

Key Concepts

Retail Price Calculation

To understand how retail price calculation works in matrix multiplication, let's start with the basics. Imagine you have a shop selling different frozen food items. Each of these items has a different price. Mathematically, we can represent these prices using a matrix called matrix \(B\). This matrix contains the retail prices of the individual items. In our example from Joe's Specialty Foods, the retail prices for pizza, ravioli, and macaroni and cheese are \(4, \)20, and $10, respectively.

The beauty of using matrices is that they simplify complex calculations like total revenue. When you multiply this price matrix \(B\) with the sales distribution matrix \(A\), you get a new matrix that tells us the revenue made each day. The multiplication involves taking each item sold on a given day and multiplying it by its retail price, then summing these products together. This mathematical operation allows us to predict daily earnings efficiently without manual calculations for each item. It's a powerful way to both handle and optimize pricing and sales data.

Sales Distribution

Sales distribution is a key aspect of managing any retail business, and it's crucial for understanding how products are performing over time. In our scenario, the distribution of sales for frozen foods over three days can be depicted using matrix \(A\).

This matrix gives us a detailed look at how many units of each food item are sold each day. For instance, matrix \(A\) tells us that on Monday, Joe's Specialty Foods sold 50 pizzas, 20 ravioli, and 15 mac and cheeses. By organizing this data into a matrix form, you can see patterns and trends more clearly. You can analyze whether one product is outperforming others or whether specific days see more sales.

When combined with retail price information, sales distribution not only aids in forecasting potential revenue but also helps in resource allocation and stock management, ensuring the right amount of products is available to meet demand on each day.

Revenue Analysis

Revenue analysis involves looking at the total income generated from sales, which is vital for assessing the financial health of a business.

By multiplying the sales distribution matrix \(A\) and the prices matrix \(B\), you calculate the product matrix \(AB\). Each element in this resulting matrix represents daily revenue obtained from sales of all products. From our example, revenue generation across Monday, Tuesday, and Wednesday yields a total of \(750, \)1860, and $2340, respectively.

Performing this type of matrix multiplication not only simplifies the calculation but also neatly organizes the revenue data, making it easier to analyze trends and predict future income. This is efficient for businesses to regularly monitor their earnings and make informed decisions based on quantifiable data. Such analysis helps businesses in strategizing for growth, understanding profit margins, and aligning their marketing efforts to boost sales further.