A juice bar offers various smoothies on its menu, including strawberry and mango. A small portion of either costs [tex]\$4.75[/tex], and a large portion costs [tex]\$5.50[/tex]. During a short period of time, the number of smoothies sold is shown in the table below:

[tex]\[
\begin{tabular}{|c|c|c|}
\hline
& Small & Large \\
\hline
Strawberry & 6 & 3 \\
\hline
Mango & 4 & 7 \\
\hline
\end{tabular}
\][/tex]

[a] Write down a column matrix [tex]N[/tex] representing the cost of each type of smoothie.

[b] Given that [tex]M=\left[\begin{array}{cc}6 & 3 \\ 4 & 7\end{array}\right][/tex], evaluate [tex]MN[/tex].

[c] Explain what the numbers given in the answer in [b] signify.



Answer :

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.