6. A linear programming problem in \(x\) and \(y\) is described as follows.
Minimise \(C = 2 x + 3 y\)
subject to
$$\begin{aligned}
x + y & \geqslant 8
x & < 8
4 y & \geqslant x
3 y & \leqslant 9 + 2 x
\end{aligned}$$
- Add lines and shading to Diagram 1 in the answer book to represent these constraints.
- Hence determine the feasible region and label it R .
- Use the objective line (ruler) method to find the exact coordinates of the optimal vertex, V, of the feasible region. You must draw and label your objective line clearly.
- Calculate the corresponding value of \(C\) at V .
The objective is now to maximise \(2 x + 3 y\), where \(x\) and \(y\) are integers.
- Write down the optimal values of \(x\) and \(y\) and the corresponding maximum value of \(2 x + 3 y\).
A further constraint, \(y \leqslant k x\), where \(k\) is a positive constant, is added to the linear programming problem.
- Determine the least value of \(k\) for which this additional constraint does not affect the feasible region.