D1 June 2014 (R) Q5

EdexcelOld spec11 marksLinear Programming

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

Maximise \(P = 2x + 3y\)

subject to

\[\begin{aligned} x &\geqslant 25\\ y &\geqslant 25\\ 7x + 8y &\leqslant 840\\ 4y &\leqslant 5x\\ 5y &\geqslant 3x\\ x,\ y &\geqslant 0 \end{aligned}\]

(a) Add lines and shading to Diagram 1 in the answer book to represent these constraints. Hence determine the feasible region and label it R. (4)

(b) Use the objective line 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)

Given that an integer solution is required,

(d) determine the optimal solution with integer coordinates. You must make your method clear. (2)

Graph of x = 25, y = 25, 7x + 8y = 840, 4y = 5x, 5y = 3x with feasible region R, objective line and vertex V — Mark scheme (a)

Scheme	Marks
	B1 B1 B1 B1 R
	(4)

Mark scheme (b)

Scheme	Marks
Drawing an objective line accept reciprocal gradient	M1
correct objective line minimum length equivalent to (0, 10) to (15,0)	A1
V labelled correctly	A1
	(3)

Notes

b1M1: Drawing the correct objective line or its reciprocal. Line must be correct to within one small square if extended from axis to axis.

b1A1: Correct objective line.

b2A1: V labelled clearly on their graph. This mark is dependent on the correct five line segments that define the boundary of the feasible region.

Mark scheme (c)

Scheme	Marks
\(\text{V}\left(49\dfrac{7}{17},\ 61\dfrac{13}{17}\right)\)	M1 A1
	(2)

Notes

cM1: Simultaneous equation being used to find their V (but not from \(x = 25\) or \(y = 25\)). Must get to \(x = \ldots\) and \(y = \ldots\)

cA1: Correct coordinates of V stated exactly as \(\left(\dfrac{840}{17},\ \dfrac{1050}{17}\right)\) or \(\left(49\dfrac{7}{17},\ 61\dfrac{13}{17}\right)\). If the correct coordinates are stated exactly with no working then this scores M1A0.

Mark scheme (d)

Scheme	Marks
Testing the correct inequalities for points with integer coordinates	M1
(50, 61)	A1
	(2)
	(11 marks)

Notes

d1M1: Testing the correct inequalities for at least three of (49, 61), (49, 62), (50, 61), (50, 62).

d1A1: CAO (50, 61).