8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{552f3296-ad61-448b-8168-6709fb359fa2-8_1051_1385_194_365}
\captionsetup{labelformat=empty}
\caption{Figure 6}
\end{figure}
A company produces two products, X and Y .
Let \(x\) and \(y\) be the hourly production, in kgs, of X and Y respectively.
In addition to \(x \geqslant 0\) and \(y \geqslant 0\), two of the constraints governing the production are
$$\begin{gathered}
12 x + 7 y \geqslant 840
4 x + 9 y \geqslant 720
\end{gathered}$$
These constraints are shown on the graph in Figure 6, where the rejected regions are shaded out. Two further constraints are
$$\begin{gathered}
x \geqslant 20
3 x + 2 y \leqslant 360
\end{gathered}$$
- Add two lines and shading to Figure 6 in your answer book to represent these inequalities.
- Hence determine and label the feasible region, R.
The company makes a profit of 70 p and 20 p per kilogram of X and Y respectively.
- Write down an expression, in terms of \(x\) and \(y\), for the hourly profit, £P.
- Mark points A and B on your graph where A and B represent the maximum and minimum values of P respectively. Make your method clear.
(4)