Problem 28 - Algebra 2

Question

Solada Fox sells fruit from her three farms. Apples are \(\$ 22\) a case, peaches are \(\$ 25\) a case, and apricots are \(\$ 18\) a case. $$ \begin{array}{|c|c|c|c|} \hline \text { Farm } & \text { Apples } & \text { Peaches } & \text { Apricots } \\ \hline 1 & 290 & 165 & 210 \\ \hline 2 & 175 & 240 & 190 \\ \hline 3 & 110 & 75 & 0 \\ \hline \end{array} $$ Write an inventory matrix for the number of cases for each type of fruit for each farm and a cost matrix for the price per case for each type of fruit.

Step-by-Step Solution

Answer

Inventory matrix: \( \begin{bmatrix} 290 & 165 & 210 \\ 175 & 240 & 190 \\ 110 & 75 & 0 \end{bmatrix} \) Cost matrix: \( \begin{bmatrix} 22 & 25 & 18 \end{bmatrix} \)

Step 1: Understand the Problem

The goal is to create two matrices: one for the inventory of fruit cases across farms and another for the costs of these fruits. The inventory matrix will show how many cases of each fruit type are produced on each farm. The cost matrix will specify the price per case for each fruit type.

Step 2: Create the Inventory Matrix

The inventory matrix represents the number of cases per fruit type on each farm. Based on the given table, we construct:\[\text{Inventory Matrix} = \begin{bmatrix}290 & 165 & 210 \175 & 240 & 190 \110 & 75 & 0\end{bmatrix}\]Here, rows represent different farms (Farm 1, Farm 2, Farm 3), and columns represent the fruit types (Apples, Peaches, Apricots).

Step 3: Construct the Cost Matrix

The cost matrix will represent the cost per case of each fruit type. As the prices are constant across all farms, this will be a single row matrix:\[\text{Cost Matrix} = \begin{bmatrix} 22 & 25 & 18 \end{bmatrix}\]This matrix has one row which corresponds to different types of fruits: Apples, Peaches, and Apricots.

Key Concepts

Understanding the Inventory Matrix

An inventory matrix is an essential tool in applied algebra, especially when dealing with large sets of data like stocks from different locations or products. In the context of Solada Fox's fruit farms, the inventory matrix provides a clear way to organize and visualize the number of fruit cases available at each farm.
Each row in the matrix represents a particular farm, allowing us to clearly see how much inventory each farm holds. The columns correspond to different fruit types. In this scenario, we have apples, peaches, and apricots as our columns.

• Apples, positioned in the first column, show how many apple cases each farm produces.
• Peaches take the second column, detailing peach production per farm.
• The third column is reserved for apricot cases.

By organizing data this way, one can quickly discern patterns or anomalies in production between different farms. For instance, Farm 3 produces no apricots, and having this data neatly consolidated into a matrix makes it easy to spot such details.

Applying Cost Matrix Concepts

The cost matrix is a concise representation showing how much each type of fruit costs. This matrix helps in calculating total expenses or revenue generated from selling these fruits. In Solada's case, each fruit type has a fixed price per case. Therefore, the cost matrix is merely a single row that lists these prices:

• 22 equates to the cost per case for apples.
• 25 reflects the price for peaches.
• 18 stands for apricots.

Simplifying the pricing structure into a matrix format allows for easy multiplication with the inventory matrix. This operation yields the total value of the inventory across all farms. It's a practical application of linear algebra that helps to manage costs, set pricing strategies, and ensure profitability.

Applied Algebra in Inventory and Costs

Applied algebra plays a crucial role in understanding and managing real-world business problems like inventories and costs efficiently. Algebra is utilized here to transform complex data sets into simple, manageable mathematical matrices.
When you use matrices to represent inventory and costs, you unlock powerful tools for analysis and forecasting. Combining the inventory matrix with the cost matrix through matrix multiplication provides a swift means to compute total sales revenue. This operation is straightforward: each element of the resultant matrix represents total earnings from selling a specific type of fruit from each farm.

• Matrix operations help calculate future projections and identify trends.
• Utilizing applied algebra simplifies decision-making processes.
• It ensures that businesses like Solada Fox's can remain competitive by quickly adapting to changes in fruit production or market prices.

Overall, this exemplifies how fundamental mathematical concepts, particularly matrices, aid in effective business management and strategic planning.