D1 June 2013 (R) Q8

EdexcelOld spec16 marksLinear Programming

8.

Figure 6: graph with the lines y = 16 and a line through the origin, rejected regions shaded
Figure 6

A company makes two types of garden bench, the ‘Rustic’ and the ‘Contemporary’. The company wishes to maximise its profit and decides to use linear programming.

Let \(x\) be the number of ‘Rustic’ benches made each week and \(y\) be the number of ‘Contemporary’ benches made each week.

The graph in Figure 6 is being used to solve this linear programming problem.

Two of the constraints have been drawn on the graph and the rejected region shaded out.

(a) Write down the constraints shown on the graph giving your answers as inequalities in terms of \(x\) and \(y\). (3)

It takes 4 working hours to make one ‘Rustic’ bench and 3 working hours to make one ‘Contemporary’ bench. There are 120 working hours available in each week.

(b) Write down an inequality to represent this information. (2)

Market research shows that ‘Rustic’ benches should be at most \(\dfrac{3}{4}\) of the total benches made each week.

(c) Write down, and simplify, an inequality to represent this information. Your inequality must have integer coefficients. (2)
(d) Add two lines and shading to Diagram 1 in your answer book to represent the inequalities of (b) and (c). Hence determine and label the feasible region, R. (3)

The profit on each ‘Rustic’ bench and each ‘Contemporary’ bench is £45 and £30 respectively.

(e) Write down the objective function, P, in terms of \(x\) and \(y\). (1)
(f) Determine the coordinates of each of the vertices of the feasible region and hence use the vertex method to determine the optimal point. (4)
(g) State the maximum weekly profit the company could make. (1)
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