Vince works on a construction site. The amount Vince gets paid depends on the type of shift he works, as shown in the table below. \begin{array}{|l|c|c|c|} This information is shown in matrix \(R\) below. \begin{align*} --- 3 WORK AREA LINES (style=lined) --- During one week, Vince works 28 hours at the normal rate of pay, 6 hours at the overtime rate of pay, and 8 hours at the weekend rate of pay. --- 0 WORK AREA LINES (style=lined) --- Vince will receive $90 per hour if he works a public holiday shift. Matrix \(Q\), as calculated below, can be used to show Vince's hourly rate for each type of shift. \begin{align*} --- 3 WORK AREA LINES (style=lined) ---
\hline
\rule{0pt}{2.5ex} \textbf{Shift type} \rule[-1ex]{0pt}{0pt}& \textbf{Normal} & \textbf{Overtime} & \textbf {Weekend} \\
\hline
\rule{0pt}{2.5ex} \textbf{Hourly rate of pay} \rule[-1ex]{0pt}{0pt} \ \text{(\$ per hour)} & 36 & 54 & 72 \\
\hline
\end{array}
R=\left[\begin{array}{lll}
36 & 54 & 72
\end{array}\right]
\end{align*}
\begin{aligned}
Q & =n \times\left[\begin{array}{llll}
1 & 1.5 & 2 & p
\end{array}\right] \\
& =\left[\begin{array}{llll}
36 & 54 & 72 & 90
\end{array}\right]
\end{aligned}
\end{align*}
MATRICES, FUR2 2019 VCAA 1
The car park at a theme park has three areas, `A, B` and `C`.
The number of empty `(E)` and full `(F)` parking spaces in each of the three areas at 1 pm on Friday are shown in matrix `Q` below.
`{:(qquad qquad qquad \ E qquad F),(Q = [(70, 50),(30, 20),(40, 40)]{:(A),(B),(C):}quad text(area)):}`
- What is the order of matrix `Q`? (1 mark)
--- 1 WORK AREA LINES (style=lined) ---
- Write down a calculation to show that 110 parking spaces are full at 1 pm. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
Drivers must pay a parking fee for each hour of parking.
Matrix `P`, below, shows the hourly fee, in dollars, for a car parked in each of the three areas.
`{:(qquad qquad qquad qquad qquad text{area}), (qquad qquad qquad A qquad quad quad B qquad qquad C), (P = [(1.30, 3.50, 1.80)]):}`
- The total parking fee, in dollars, collected from these 110 parked cars if they were parked for one hour is calculated as follows.
`qquad qquad qquad P xx L = [207.00]`
where matrix `L` is a `3 xx 1` matrix.
Write down matrix `L`. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
The number of whole hours that each of the 110 cars had been parked was recorded at 1 pm. Matrix `R`, below, shows the number of cars parked for one, two, three or four hours in each of the areas `A, B` and `C`.
`{:(qquadqquadqquadqquadquadtext(area)),(quad qquadqquadquad \ A qquad B qquad C),(R = [(3, 1, 1),(6, 10, 3),(22, 7,10),(19, 2, 26)]{:(1),(2),(3),(4):}\ text(hours)):}`
- Matrix `R^T` is the transpose of matrix `R`.
Complete the matrix `R^T` below. (1 mark)
--- 0 WORK AREA LINES (style=lined) ---
`qquad R^T = [( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , )]`
- Explain what the element in row 3, column 2 of matrix `R^T` represents. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---