D1 January 2008 Q7

EdexcelOld spec16 marksLinear Programming

7.

Figure 7: graph with x from 0 to 160 and y from 0 to 100 showing the line x + y = 70 and a fourth line through the origin, rejected region shaded
Figure 7

Phil sells boxed lunches to travellers at railway stations. Customers can select either the vegetarian box or the non-vegetarian box.

Phil decides to use graphical linear programming to help him optimise the numbers of each type of box he should produce each day.

Each day Phil produces \(x\) vegetarian boxes and \(y\) non-vegetarian boxes.

One of the constraints limiting the number of boxes is

\[x + y \geqslant 70.\]

This, together with \(x \geqslant 0\), \(y \geqslant 0\) and a fourth constraint, has been represented in Figure 7. The rejected region has been shaded.

(a) Write down the inequality represented by the fourth constraint. (2)

Two further constraints are:

\[x + 2y \leqslant 160\]\[\text{and} \quad y \gt 60.\]
(b) Add two lines and shading to Diagram 4 in your answer book to represent these inequalities. (4)
(c) Hence determine and label the feasible region, R. (1)
(d) Use your graph to determine the minimum total number of boxes he needs to prepare each day. Make your method clear. (3)

Phil makes a profit of £1.20 on each vegetarian box and £1.40 on each non-vegetarian box. He wishes to maximise his profit.

(e) Write down the objective function. (1)
(f) Use your graph to obtain the optimal number of vegetarian and non-vegetarian boxes he should produce each day. You must make your method clear. (4)
(g) Find Phil’s maximum daily profit. (1)
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