A company sells chocolate muffins and banana muffins. These muffins are sold across three different outlets, which are Outlet A, Outlet B and Outlet C. The table shows the number of boxes of chocolate muffins and banana muffins produced at each of the three outlets in a particular day. The table can be represented by matrix Q, where $Q=\begin{pmatrix}36 & 40\\48 & 39\\45 & x\end{pmatrix}$

1. Write the two production costs as a $2\times 1$ column matrix in the correct order.

The columns of $Q$ are in the order: Chocolate Muffin, Banana Muffin.

So the cost matrix is
$P=\begin{pmatrix}3\\4.50\end{pmatrix}.$

1. Carry out row-by-column multiplication to find each element of $R=QP$.

$R=QP=\begin{pmatrix}36&40\\48&39\\45&x\end{pmatrix}\begin{pmatrix}3\\4.50\end{pmatrix}$

$=\begin{pmatrix}36(3)+40(4.50)\\48(3)+39(4.50)\\45(3)+x(4.50)\end{pmatrix}$

$=\begin{pmatrix}108+180\\144+175.50\\135+4.50x\end{pmatrix}$

$=\begin{pmatrix}288\\319.50\\135+4.50x\end{pmatrix}$

1. Identify what the matrix multiplication $QP$ calculates.

Matrix $Q$ gives the number of boxes of each flavour produced at each outlet, and matrix $P$ gives the production cost per box of each flavour.

Thus each element of $R=QP$ is:
$\text{number of chocolate boxes}\times 3+\text{number of banana boxes}\times 4.50$

2. State the contextual meaning of the elements of $R$.

Since the rows of $Q$ correspond to Outlet A, Outlet B and Outlet C, the elements of $R$ represent the total production costs for Outlet A, Outlet B and Outlet C respectively.

1. Convert the selling price condition into a multiplier.

Selling price is $70\%$ more than production cost.

$100\%+70\%=170\%=1.7$

2. Use the entries of $R$ with the fractions sold to find the production cost of the muffins sold at each outlet.

From part (a)(ii),

$R=\begin{pmatrix}288\\319.50\\135+4.50x\end{pmatrix}$

Production cost of muffins sold:

Outlet A:
$\frac{5}{6}(288)=240$

Outlet B:
$\frac{4}{5}(319.50)=255.60$

Outlet C:
$1(135+4.50x)=135+4.50x$

3. Multiply the total production cost of muffins sold by $1.7$ and add to get the total amount collected.

Total amount collected:

$1.7\left(240+255.60+135+4.50x\right)$

$=1.7(630.60+4.50x)$

$=1072.02+7.65x$

1. Form and solve the linear equation using the total amount collected.

From part (c), the total amount collected is
$(1072.02+7.65x)\text{ dollars}.$
Given that this is $1454.52,
$1072.02+7.65x=1454.52$
$7.65x=1454.52-1072.02=382.50$
$x=\frac{382.50}{7.65}=50$