Phil is to buy some squash balls for his club. There are three different types of ball that he can buy: slow, medium and fast.

He must buy at least 190 slow balls, at least 50 medium balls and at least 50 fast balls.

He must buy at least 300 balls in total.

Each slow ball costs £2.50, each medium ball costs £2.00 and each fast ball costs £2.00.

He must spend no more than £1000 in total.

At least 60% of the balls that he buys must be slow balls.

Phil buys $x$ slow balls, $y$ medium balls and $z$ fast balls.

\begin{enumerate}[label=(\alph*)]
\item Find six inequalities that model Phil's situation. [4 marks]

\item Phil decides to buy the same number of medium balls as fast balls.
\begin{enumerate}[label=(\roman*)]
\item Show that the inequalities found in part (a) simplify to give
$$x \geq 190, \quad y \geq 50, \quad x + 2y \geq 300, \quad 5x + 8y \leq 2000, \quad y \leq \frac{1}{3}x$$ [2 marks]

\item Phil sells all the balls that he buys to members of the club. He sells each slow ball for £3.00, each medium ball for £2.25 and each fast ball for £2.25. He wishes to maximise his profit.

On Figure 1 on page 14, draw a diagram to enable this problem to be solved graphically, indicating the feasible region and the direction of an objective line. [7 marks]

\item Find Phil's maximum possible profit and state the number of each type of ball that he must buy to obtain this maximum profit. [4 marks]
\end{enumerate}
\end{enumerate}

\hfill \mbox{\textit{AQA D1 2010 Q6 [17]}}