Sure, let's address each part of the question step-by-step.
### [a] Writing down the column matrix [tex]\( N \)[/tex] representing the cost of each type of smoothie:
The costs for the smoothies are given as:
- Small portion costs \[tex]$4.75
- Large portion costs \$[/tex]5.50
We can represent these costs in a column matrix [tex]\( N \)[/tex]:
[tex]\[
N = \begin{pmatrix}
4.75 \\
5.50
\end{pmatrix}
\][/tex]
### [b] Given matrix [tex]\( M = \begin{pmatrix}
6 & 3 \\
4 & 7
\end{pmatrix} \)[/tex], evaluate [tex]\( MN \)[/tex]:
First, let's re-write the given matrices [tex]\( M \)[/tex] and [tex]\( N \)[/tex]:
[tex]\( M = \begin{pmatrix}
6 & 3 \\
4 & 7
\end{pmatrix} \)[/tex]
[tex]\( N = \begin{pmatrix}
4.75 \\
5.50
\end{pmatrix} \)[/tex]
Now, to find [tex]\( MN \)[/tex], we need to perform matrix multiplication:
[tex]\[
MN = \begin{pmatrix}
6 & 3 \\
4 & 7
\end{pmatrix}
\begin{pmatrix}
4.75 \\
5.50
\end{pmatrix}
\][/tex]
To perform the multiplication, multiply each element of the rows of [tex]\( M \)[/tex] by the corresponding element of the column [tex]\( N \)[/tex] and then sum these products for each entry in the resulting matrix:
[tex]\[
\begin{aligned}
MN &= \begin{pmatrix}
(6 \times 4.75) + (3 \times 5.50) \\
(4 \times 4.75) + (7 \times 5.50)
\end{pmatrix} \\
&= \begin{pmatrix}
28.5 + 16.5 \\
19 + 38.5
\end{pmatrix} \\
&= \begin{pmatrix}
45.0 \\
57.5
\end{pmatrix}
\][/tex]
### [c] Explanation of the numbers in the answer to [b]:
The numbers in the resulting matrix from part [b] [tex]\( MN \)[/tex] represent the total sales revenue for each type of smoothie.
- The first element (45.0) represents the total revenue generated from the sales of strawberry smoothies.
- The second element (57.5) represents the total revenue generated from the sales of mango smoothies.
By breaking it down:
- For strawberry smoothies: [tex]\( 6 \times \$4.75 \)[/tex] from small portions plus [tex]\( 3 \times \$5.50 \)[/tex] from large portions give a total revenue of \[tex]$45.0.
- For mango smoothies: \( 4 \times \$[/tex]4.75 \) from small portions plus [tex]\( 7 \times \$5.50 \)[/tex] from large portions give a total revenue of \[tex]$57.5.
So, we find that the total revenue from strawberry smoothies is \$[/tex]45.0 and from mango smoothies is \$57.5.
