一家公司销售巧克力马芬和香蕉马芬。这些马芬在三个不同的分店（分店 A、分店 B 和分店 C）销售。下表显示了某一天三个分店中每个分店生产的巧克力马芬和香蕉马芬的箱数。 该表可以用矩阵 Q 表示，其中 $Q=\begin{pmatrix}36 & 40\\48 & 39\\45 & x\end{pmatrix}$

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,
$1072.02+7.65x=1454.52$
$7.65x=1454.52-1072.02=382.50$
$x=\frac{382.50}{7.65}=50$