8. A chemical company produces two products $X$ and $Y$. Based on potential demand, the total production each week must be at least 380 gallons. A major customer's weekly order for 125 gallons of $Y$ must be satisfied.

Product $X$ requires 2 hours of processing time for each gallon and product $Y$ requires 4 hours of processing time for each gallon. There are 1200 hours of processing time available each week. Let $x$ be the number of gallons of $X$ produced and $y$ be the number of gallons of $Y$ produced each week.
\begin{enumerate}[label=(\alph*)]
\item Write down the inequalities that $x$ and $y$ must satisfy.\\
(3)

It costs $\pounds 3$ to produce 1 gallon of $X$ and $\pounds 2$ to produce 1 gallon of $Y$. Given that the total cost of production is $\pounds C$,
\item express $C$ in terms of $x$ and $y$.\\
(1)

The company wishes to minimise the total cost.
\item Using the graphical method, solve the resulting Linear Programming problem. Find the optimal values of $x$ and $y$ and the resulting total cost.
\item Find the maximum cost of production for all possible choices of $x$ and $y$ which satisfy the inequalities you wrote down in part (a).
\end{enumerate}

\hfill \mbox{\textit{Edexcel D1 2002 Q8 [14]}}