6. A linear programming problem in \(x\) and \(y\) is described as follows. Maximise \(P = k x + y\), where \(k\) is a constant
subject to: \(\quad 3 y \geqslant x\) $$\begin{aligned} x + 2 y & \leqslant 130
4 x + y & \geqslant 100
4 x + 3 y & \leqslant 300 \end{aligned}$$

1. Add lines and shading to Diagram 1 in the answer book to represent these constraints. Hence determine the feasible region and label it \(R\).
2. For the case when \(k = 0.8\)
   1. use the objective line method to find the optimal vertex, \(V\), of the feasible region. You must draw and label your objective line and label vertex \(V\) clearly.
   2. calculate the coordinates of \(V\) and hence calculate the corresponding value of \(P\) at \(V\). Given that for a different value of \(k , V\) is not the optimal vertex of \(R\),
3. determine the range of possible values for \(k\). You must make your method and working clear.