Answer :
Sure! Let's break down the problem and solve it step-by-step.
### Part A
To create matrix [tex]\( A \)[/tex] which represents the number of jars of strawberry and raspberry jam sold through each outlet, we need to structure it as a [tex]\( 3 \times 2 \)[/tex] matrix. Here are the given quantities:
- Fruity: 150 jars of strawberry, 100 jars of raspberry
- Jammy: 200 jars of strawberry, 75 jars of raspberry
- Nature: 50 jars of strawberry, 25 jars of raspberry
We'll arrange these values into matrix [tex]\( A \)[/tex]:
[tex]\[ A = \begin{bmatrix} 150 & 100 \\ 200 & 75 \\ 50 & 25 \\ \end{bmatrix} \][/tex]
### Part B
Next, we need to create matrix [tex]\( B \)[/tex] to represent the profit per jar for each type of jam, structured as a [tex]\( 2 \times 1 \)[/tex] matrix. The given profits are:
- Strawberry jam: \[tex]$1.50 per jar - Raspberry jam: \$[/tex]2.00 per jar
We'll arrange these values into matrix [tex]\( B \)[/tex]:
[tex]\[ B = \begin{bmatrix} 1.50 \\ 2.00 \\ \end{bmatrix} \][/tex]
### Calculating Total Profit for Each Shop
To find the total profit for each shop, we need to perform matrix multiplication between matrix [tex]\( A \)[/tex] (representing the number of jars sold) and matrix [tex]\( B \)[/tex] (representing the profit per jar).
Given:
[tex]\[ A = \begin{bmatrix} 150 & 100 \\ 200 & 75 \\ 50 & 25 \\ \end{bmatrix} \][/tex]
[tex]\[ B = \begin{bmatrix} 1.50 \\ 2.00 \\ \end{bmatrix} \][/tex]
The result of the matrix multiplication [tex]\( A \times B \)[/tex] will give us a [tex]\( 3 \times 1 \)[/tex] matrix representing the total profit for each shop:
[tex]\[ Total\_Profit = A \times B = \begin{bmatrix} 150 \times 1.50 + 100 \times 2.00 \\ 200 \times 1.50 + 75 \times 2.00 \\ 50 \times 1.50 + 25 \times 2.00 \\ \end{bmatrix} \][/tex]
Calculating each element:
- For Fruity:
[tex]\[ 150 \times 1.50 + 100 \times 2.00 = 225 + 200 = 425 \][/tex]
- For Jammy:
[tex]\[ 200 \times 1.50 + 75 \times 2.00 = 300 + 150 = 450 \][/tex]
- For Nature:
[tex]\[ 50 \times 1.50 + 25 \times 2.00 = 75 + 50 = 125 \][/tex]
So, the total profit matrix is:
[tex]\[ Total\_Profit = \begin{bmatrix} 425.00 \\ 450.00 \\ 125.00 \\ \end{bmatrix} \][/tex]
### Summary
- Matrix [tex]\( A \)[/tex] (Number of jars sold):
[tex]\[ \begin{bmatrix} 150 & 100 \\ 200 & 75 \\ 50 & 25 \\ \end{bmatrix} \][/tex]
- Matrix [tex]\( B \)[/tex] (Profit per jar):
[tex]\[ \begin{bmatrix} 1.50 \\ 2.00 \\ \end{bmatrix} \][/tex]
- Resulting Total Profit Matrix:
[tex]\[ \begin{bmatrix} 425.00 \\ 450.00 \\ 125.00 \\ \end{bmatrix} \][/tex]
This step-by-step breakdown demonstrates how we can use matrix multiplication to solve the problem of calculating total profits based on the number of jars sold and the profit per jar.
### Part A
To create matrix [tex]\( A \)[/tex] which represents the number of jars of strawberry and raspberry jam sold through each outlet, we need to structure it as a [tex]\( 3 \times 2 \)[/tex] matrix. Here are the given quantities:
- Fruity: 150 jars of strawberry, 100 jars of raspberry
- Jammy: 200 jars of strawberry, 75 jars of raspberry
- Nature: 50 jars of strawberry, 25 jars of raspberry
We'll arrange these values into matrix [tex]\( A \)[/tex]:
[tex]\[ A = \begin{bmatrix} 150 & 100 \\ 200 & 75 \\ 50 & 25 \\ \end{bmatrix} \][/tex]
### Part B
Next, we need to create matrix [tex]\( B \)[/tex] to represent the profit per jar for each type of jam, structured as a [tex]\( 2 \times 1 \)[/tex] matrix. The given profits are:
- Strawberry jam: \[tex]$1.50 per jar - Raspberry jam: \$[/tex]2.00 per jar
We'll arrange these values into matrix [tex]\( B \)[/tex]:
[tex]\[ B = \begin{bmatrix} 1.50 \\ 2.00 \\ \end{bmatrix} \][/tex]
### Calculating Total Profit for Each Shop
To find the total profit for each shop, we need to perform matrix multiplication between matrix [tex]\( A \)[/tex] (representing the number of jars sold) and matrix [tex]\( B \)[/tex] (representing the profit per jar).
Given:
[tex]\[ A = \begin{bmatrix} 150 & 100 \\ 200 & 75 \\ 50 & 25 \\ \end{bmatrix} \][/tex]
[tex]\[ B = \begin{bmatrix} 1.50 \\ 2.00 \\ \end{bmatrix} \][/tex]
The result of the matrix multiplication [tex]\( A \times B \)[/tex] will give us a [tex]\( 3 \times 1 \)[/tex] matrix representing the total profit for each shop:
[tex]\[ Total\_Profit = A \times B = \begin{bmatrix} 150 \times 1.50 + 100 \times 2.00 \\ 200 \times 1.50 + 75 \times 2.00 \\ 50 \times 1.50 + 25 \times 2.00 \\ \end{bmatrix} \][/tex]
Calculating each element:
- For Fruity:
[tex]\[ 150 \times 1.50 + 100 \times 2.00 = 225 + 200 = 425 \][/tex]
- For Jammy:
[tex]\[ 200 \times 1.50 + 75 \times 2.00 = 300 + 150 = 450 \][/tex]
- For Nature:
[tex]\[ 50 \times 1.50 + 25 \times 2.00 = 75 + 50 = 125 \][/tex]
So, the total profit matrix is:
[tex]\[ Total\_Profit = \begin{bmatrix} 425.00 \\ 450.00 \\ 125.00 \\ \end{bmatrix} \][/tex]
### Summary
- Matrix [tex]\( A \)[/tex] (Number of jars sold):
[tex]\[ \begin{bmatrix} 150 & 100 \\ 200 & 75 \\ 50 & 25 \\ \end{bmatrix} \][/tex]
- Matrix [tex]\( B \)[/tex] (Profit per jar):
[tex]\[ \begin{bmatrix} 1.50 \\ 2.00 \\ \end{bmatrix} \][/tex]
- Resulting Total Profit Matrix:
[tex]\[ \begin{bmatrix} 425.00 \\ 450.00 \\ 125.00 \\ \end{bmatrix} \][/tex]
This step-by-step breakdown demonstrates how we can use matrix multiplication to solve the problem of calculating total profits based on the number of jars sold and the profit per jar.
