An online shopping platform, Maobao, ships two types of parcels: Standard and Express.
They utilise two courier services: Speedy (\(S\)) and Timely (\(T\)).
The shipping costs (in dollars) per parcel for each courier are represented by the cost matrix \(\mathbf C\).
\(\mathbf C=\begin{array}{c}\begin{array}{cc}S&T\end{array}\\\left(\begin{array}{cc}5&6\\12&x\end{array}\right)\end{array}\begin{array}{l}\phantom{S}\\\text{Standard}\\\text{Express}\end{array}\)
Maobao has a shipment of 80 Standard parcels and 50 Express parcels on Monday and a shipment of 100 Standard parcels and 40 Express parcels on Tuesday.
Write down a \(2\times2\) matrix, \(\mathbf A\), to represent the parcel quantities, where the rows represent the days (Monday and Tuesday respectively) and the columns represent the parcel types (Standard and Express respectively).[1]
Find, in terms of \(x\), the matrix \(\mathbf M=\mathbf{AC}\).[2]
Explain what the element in the second row and first column of your answer in part (b) represents.[1]
The elements of the matrix \(\mathbf B\), where \(\mathbf B=\mathbf{MD}\), represent the cost differences of the two courier services for Monday and Tuesday respectively.
Write down the matrix \(\mathbf D\).[1]
Find, in terms of \(x\), the matrix \(\mathbf B=\mathbf{MD}\).[1]
The cost of courier service for Tuesday is \(\$20\) cheaper if Maobao engage Speedy’s courier service.
Find the value of \(x\).[2]
The operations manager states:
“Our total shipping bill will be exactly the same for the two days for either courier services.”
Justify whether the manager’s claim is correct.[1]