Problem

Figure 3 shows the constraints of a linear programming problem in $x$ and $y$.

The unshaded area, including its boundaries, forms the feasible region, $R$.

The four vertices of $R$ are $A(6, 8)$, $B(13, 12)$, $C(9, 22)$ and $D(5, 18)$.

An objective line has been drawn and labelled on the graph.

When the objective function, $P$, is maximised, the value of $P$ is 540.

When the objective function, $P$, is minimised, the value of $P$ is $k$.

Determine the value of $k$. You must make your method and working clear.

(You may assume that the objective function, $P$, takes the form $ax + by$ where $a$ and $b$ are constants.)

(Total for Question 5 is 5 marks)