7. A tailor makes two types of garment, $A$ and $B$. He has available $70 \mathrm {~m} ^ { 2 }$ of cotton fabric and $90 \mathrm {~m} ^ { 2 }$ of woollen fabric. Garment $A$ requires $1 \mathrm {~m} ^ { 2 }$ of cotton fabric and $3 \mathrm {~m} ^ { 2 }$ of woollen fabric. Garment $B$ requires $2 \mathrm {~m} ^ { 2 }$ of each fabric.

The tailor makes $x$ garments of type $A$ and $y$ garments of type $B$.
\begin{enumerate}[label=(\alph*)]
\item Explain why this can be modelled by the inequalities

$$\begin{aligned}
& x + 2 y \leq 70 \\
& 3 x + 2 y \leq 90 \\
& x \geq 0 , y \geq 0
\end{aligned}$$

(2 marks)\\
The tailor sells type $A$ for $\pounds 30$ and type $B$ for $\pounds 40$. All garments made are sold. The tailor wishes to maximise his total income.
\item Set up an initial Simplex tableau for this problem.\\
(3 marks)
\item Solve the problem using the Simplex algorithm.\\
(8 marks)

Figure 4 shows a graphical representation of the feasible region for this problem.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{3147dad8-2d3c-42fd-b288-7017ff1fce16-004_452_828_995_356}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{center}
\end{figure}
\item Obtain the coordinates of the points A, $C$ and $D$.
\item Relate each stage of the Simplex algorithm to the corresponding point in Fig. 4.\\
(3 marks)

Answer Book (AB12)\\
Graph Paper (ASG2)

Items included with question papers Answer booklet
\end{enumerate}

\hfill \mbox{\textit{Edexcel D1  Q7}}