3 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries.
\includegraphics[max width=\textwidth, alt={}, center]{7ca6d572-d776-4ad7-a0ed-9ec43c975585-03_908_915_392_614}

1. Write down the inequalities that define the feasible region. The objective is to maximise \(P _ { 1 } = x + 6 y\).
2. Find the values of \(x\) and \(y\) at the optimal point, and the corresponding value of \(P _ { 1 }\). The objective is changed to maximise \(P _ { k } = k x + 6 y\), where \(k\) is positive.
3. Calculate the coordinates of the optimal point, and the corresponding value of \(P _ { k }\) when the optimal point is not the same as in part (ii).
4. Find the range of values of \(k\) for which the point identified in part (ii) is still optimal.