Problem

The graph in Figure 4 is being used to solve a linear programming problem in $x$ and $y$. The three constraints have been drawn on the graph and the rejected regions have been shaded out. The three vertices of the feasible region $R$ are labelled $A$, $B$ and $C$.

(a) Determine the inequalities that define $R$.

The objective function, $P$, is given by $P = ax + by$

where $a$ and $b$ are positive constants.

The minimum value of $P$ is 8 and the maximum value of $P$ occurs at $C$.

(b) Find the range of possible values of $a$. You must make your method clear.