Problem 29 - 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. Find the total income of the three fruit farms expressed as a matrix. $$\begin{array}{|c|c|c|c|} \hline \text { Farm } & \text { Apples } & \text { Peaches } & \text { Apricots } \\ \hline 1 & 290 & 165 & 210 \\ \hline 2 & 175 & 240 & 190 \\ \hline 3 & 110 & 75 & 0 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The total income matrix is \( I = \begin{bmatrix} 14285 \\ 13270 \\ 4295 \end{bmatrix} \).

Step 1: Set up Price Matrix

The prices for each type of fruit are given: Apples at \\(22 per case, Peaches at \\)25 per case, and Apricots at \$18 per case. We can represent this as a matrix: \( P = \begin{bmatrix} 22 & 25 & 18 \end{bmatrix} \).

Step 2: Set up Quantity Matrix

The quantity of fruit cases sold at each farm can be represented as a matrix based on the given data: \( Q = \begin{bmatrix} 290 & 165 & 210 \ 175 & 240 & 190 \ 110 & 75 & 0 \end{bmatrix} \). Each row represents a farm, and each column corresponds to apples, peaches, and apricots.

Step 3: Multiply Matrices

To find the total income for each farm, multiply the price matrix \( P \) with the quantity matrix \( Q \). This involves performing matrix multiplication, where the resulting matrix will have its elements calculated by the sum of products of corresponding elements from \( P \) and the rows of \( Q \).

Step 4: Calculate Income for Each Farm

First, multiply the price matrix by the first row of the quantity matrix to find the income for Farm 1: \( 22 \times 290 + 25 \times 165 + 18 \times 210 = 6380 + 4125 + 3780 = 14285 \). Do the same for the remaining farms. For Farm 2: \( 22 \times 175 + 25 \times 240 + 18 \times 190 = 3850 + 6000 + 3420 = 13270 \). For Farm 3: \( 22 \times 110 + 25 \times 75 + 18 \times 0 = 2420 + 1875 + 0 = 4295 \).

Step 5: Construct the Income Matrix

Combine the calculated incomes into a final income matrix: \( I = \begin{bmatrix} 14285 \ 13270 \ 4295 \end{bmatrix} \), which represents the income for Farms 1, 2, and 3, respectively.

Key Concepts

Price Matrix

When you're dealing with commodities like fruits that have different costs, it's useful to consolidate all price details into one concise format: the Price Matrix. In this context, each fruit is associated with a specific cost per case.
This financial detail is expressed as a single row matrix, which makes it easy for calculations because you only have one row with multiple columns, each representing a fruit.

• Apples cost \(22\), peaches cost \(\\(25\), and apricots cost \(\\)18\) per case.
• The resulting Price Matrix is: \[P = \begin{bmatrix} 22 & 25 & 18 \end{bmatrix}\]

This matrix is efficient for use in further operations, such as multiplying with another matrix.

Quantity Matrix

A Quantity Matrix details how many units of each item are involved in a particular operation or scenario. For Solada Fox's fruit business, the Quantity Matrix describes how many cases of each type of fruit were sold at the different farms.
Each row in this matrix corresponds to a specific farm, and each column represents a fruit.

• Row 1 shows cases for Farm 1: apples, peaches, and apricots.
• Row 2 illustrates counts for Farm 2, and so on.
• The complete Quantity Matrix appears as:\[Q = \begin{bmatrix} 290 & 165 & 210 \ 175 & 240 & 190 \ 110 & 75 & 0 \end{bmatrix}\]

Understanding this matrix helps in keeping track of inventory and feeds into predicting future needs or profits.

Income Matrix

An Income Matrix is a powerful tool that results from combining Price and Quantity Matrices to determine financial outcomes.
In Solada Fox's scenario, each element in the Income Matrix corresponds to the total income generated by a farm.

• To find the income from Farm 1, you multiply each fruit's quantity by its price as specified in the Price Matrix (and sum these products).
• Similarly, this is done for Farms 2 and 3.
• The results populate the Income Matrix, which looks like this:\[I = \begin{bmatrix} 14285 \ 13270 \ 4295 \end{bmatrix}\]

Each row of the Income Matrix gives a clear picture of each farm's total revenue.

Matrix Operations

Matrix Operations allow for complex calculations involving multiple datasets, like prices and quantities of goods. When matrices align in terms of dimensions, you can perform various operations like addition, subtraction, and crucially, multiplication.
In this context, we're focused on matrix multiplication – a way to combine a Price Matrix with a Quantity Matrix.

• Multiplication is achieved by taking a row from the Price Matrix and a column from the other matrix (or rows, in this case) and calculating the dot product.
• Each element from the Price Matrix is multiplied by its corresponding element from the Quantity Matrix's row, and the products are summed.
• This results in new matrices like the Income Matrix that we used to express each farm's earnings.

These operations make matrices a powerful tool in fields ranging from economics to physics.