Problem

A linear programming problem in $x$ and $y$ is described as follows.

Maximise $P = kx + y$, where $k$ is a constant

subject to:

$3y \geq x$ $x + 2y \leq 130$ $4x + y \geq 100$ $4x + 3y \leq 300$

(a) Add lines and shading to Diagram 1 in the answer book to represent these constraints. Hence determine the feasible region and label it $R$.

(b) For the case when $k = 0.8$

(i) 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.

(ii) 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 $R$,

(c) determine the range of possible values for $k$. You must make your method and working clear.