D1 June 2015 Q6

EdexcelOld spec13 marksLinear Programming

6. A linear programming problem in \(x\) and \(y\) is described as follows.

Minimise \(C = 2x + 3y\)

subject to

\[\begin{aligned} x + y &\geqslant 8\\ x &\lt 8\\ 4y &\geqslant x\\ 3y &\leqslant 9 + 2x \end{aligned}\]
(a) Add lines and shading to Diagram 1 in the answer book to represent these constraints. (4)
(b) Hence determine the feasible region and label it R. (1)
(c) 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. (3)
(d) Calculate the corresponding value of \(C\) at V. (1)

The objective is now to maximise \(2x + 3y\), where \(x\) and \(y\) are integers.

(e) Write down the optimal values of \(x\) and \(y\) and the corresponding maximum value of \(2x + 3y\). (2)

A further constraint, \(y \leqslant kx\), where \(k\) is a positive constant, is added to the linear programming problem.

(f) Determine the least value of \(k\) for which this additional constraint does not affect the feasible region. (2)
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