D1 January 2012 Q6

EdexcelOld spec11 marksLinear Programming

6.

Figure 6: axes x from 0 to 130 and y from 0 to 100; lines x = 40 and y = 50; regions x < 40 and y > 50 shaded
Figure 6

Edgar has recently bought a field in which he intends to plant apple trees and plum trees.

He can use linear programming to determine the number of each type of tree he should plant.

Let \(x\) be the number of apple trees he plants and \(y\) be the number of plum trees he plants.

Two of the constraints are

\[\begin{aligned}x &\geqslant 40\\ y &\leqslant 50\end{aligned}\]

These are shown on the graph in Figure 6, where the rejected region is shaded out.

(a) Use these two constraints to write down two statements that describe the number of apple trees and plum trees Edgar can plant. (1)

Two further constraints are

\[\begin{aligned}3x + 4y &\leqslant 360\\ x &\leqslant 2y\end{aligned}\]
(b) Add two lines and shading to Diagram 1 in your answer book to represent these inequalities. Hence determine the feasible region and label it R. (4)

Edgar will make a profit of £60 from each apple tree and £20 from each plum tree. He wishes to maximise his profit, P.

(c) Write down the objective function. (1)
(d) Use an objective line to determine the optimal point of the feasible region, R. You must make your method clear. (4)
(e) Find Edgar’s maximum profit. (1)
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