4 [Figure 2, printed on the insert, is provided for use in this question.]\\
Each year, farmer Giles buys some goats, pigs and sheep.\\
He must buy at least 110 animals.\\
He must buy at least as many pigs as goats.\\
The total of the number of pigs and the number of sheep that he buys must not be greater than 150 .\\
Each goat costs $\pounds 16$, each pig costs $\pounds 8$ and each sheep costs $\pounds 24$.\\
He has $\pounds 3120$ to spend on the animals.\\
At the end of the year, Giles sells all of the animals. He makes a profit of $\pounds 70$ on each goat, $\pounds 30$ on each pig and $\pounds 50$ on each sheep. Giles wishes to maximize his total profit, $\pounds P$.

Each year, Giles buys $x$ goats, $y$ pigs and $z$ sheep.
\begin{enumerate}[label=(\alph*)]
\item Formulate Giles's situation as a linear programming problem.
\item One year, Giles buys 30 sheep.
\begin{enumerate}[label=(\roman*)]
\item Show that the constraints for Giles's situation for this year can be modelled by

$$y \geqslant x , \quad 2 x + y \leqslant 300 , \quad x + y \geqslant 80 , \quad y \leqslant 120$$

(2 marks)
\item On Figure 2, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region and the direction of the objective line.\\
(8 marks)
\item Find Giles's maximum profit for this year and the number of each animal that he must buy to obtain this maximum profit.\\
(3 marks)
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA D1 2009 Q4 [18]}}