D1 June 2012 Q7

EdexcelOld spec13 marksLinear Programming

7.

Figure 6: graph showing the lines x = 20, y = 8 and a line labelled (a) through the origin, with shading
Figure 6

A company is going to hire out two types of car, standard and luxury.

Let \(x\) be the number of standard cars it should buy.
Let \(y\) be the number of luxury cars it should buy.

Figure 6 shows three constraints, other than \(x,\ y \geqslant 0\)
Two of these are \(x \geqslant 20\) and \(y \geqslant 8\)

(a) Write, as an inequality, the third constraint shown in Figure 6. (1)

The company decides that at least \(\dfrac{1}{6}\) of the cars must be luxury cars.

(b) Express this information as an inequality and show that it simplifies to\[5y \geqslant x\]You must make the steps in your working clear. (2)

Each time the cars are hired they need to be prepared. It takes 5 hours to prepare a standard car and it takes 6 hours to prepare a luxury car. There are 300 hours available each week to prepare the cars.

(c) Express this information as an inequality. (1)
(d) Add two lines and shading to Diagram 1 in the answer book to illustrate the constraints found in parts (b) and (c). (2)
(e) Hence determine the feasible region and label it \(R\). (1)

The company expects to make £80 profit per week on each car.

It therefore wishes to maximise \(P = 80x + 80y\), where \(P\) is the profit per week.

(f) Use the objective line (ruler) method to find the optimal vertex, \(V\), of the feasible region.
You must clearly draw and label your objective line and the vertex \(V\). (3)
(g) Given that \(P\) is the expected profit, in pounds, per week, find the number of each type of car that the company should buy and the maximum expected profit. (3)
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