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: $\quad 3 y \geqslant x$

$$\begin{aligned}
x + 2 y & \leqslant 130 \\
4 x + y & \geqslant 100 \\
4 x + 3 y & \leqslant 300
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Add lines and shading to Diagram 1 in the answer book to represent these constraints. Hence determine the feasible region and label it $R$.
\item For the case when $k = 0.8$
\begin{enumerate}[label=(\roman*)]
\item 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.
\item 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 $R$,
\end{enumerate}\item determine the range of possible values for $k$. You must make your method and working clear.
\end{enumerate}

\hfill \mbox{\textit{Edexcel D1 2021 Q6 [13]}}