Graphical solution of linear programsEdexcel International A Level Maths: Subtopic test

1. 1
   A furniture company makes $x$ desks and $y$ chairs each day. The company wants to maximise its daily profit $P=5x+4y$ (in hundreds of pounds), subject to the constraints $x+y\le10$ (worker hours), $3x+y\le18$ (workshop space), $x\ge0$ and $y\ge0$.

   (a)

   Find the coordinates of the point where the lines $x+y=10$ and $3x+y=18$ meet.

   [1 mark]

   • A$(6,4)$
   • B$(5,5)$
   • C$(3,7)$
   • D$(4,6)$

   (b)

   Which describes the ruler method for finding the optimal point?

   [1 mark]

   • ASlide a ruler parallel to the objective line away from the origin; the optimum is the last point of the feasible region it touches
   • BSlide a ruler parallel to the objective line towards the origin; the optimum is the first point of the region it touches
   • CSlide a ruler perpendicular to the objective line to the nearest vertex
   • DSlide a ruler parallel to the objective line away from the origin; the optimum is the first point of the region it touches

   (c)

   The feasible region has vertices $(0,0)$, $(6,0)$, $(4,6)$ and $(0,10)$. Use the vertex method to find the maximum value of $P$ and the values of $x$ and $y$ at which it occurs.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A school canteen mixes $x$ kg of ingredient X and $y$ kg of ingredient Y in each batch. The cost is $C=4x+3y$ pence, which is to be minimised, subject to $2x+y\ge12$ (energy), $x+2y\ge9$ (protein), $x+y\le10$ (tank capacity), $x\ge0$ and $y\ge0$.

   (a)

   Which of these points lies in the feasible region?

   [1 mark]

   • A$(3,3)$
   • B$(1,9)$
   • C$(4,4)$
   • D$(6,6)$

   (b)

   Which describes the ruler method for this minimisation problem?

   [1 mark]

   • ASlide a ruler parallel to the objective line away from the origin; the optimum is the last point of the region it touches
   • BSlide a ruler parallel to the objective line away from the origin; the optimum is the first point of the region it touches
   • CSlide a ruler perpendicular to the objective line to the nearest vertex
   • DSlide a ruler parallel to the objective line towards the origin from the far side; the optimum is the first vertex it touches

   (c)

   Show that the lines $2x+y=12$ and $x+2y=9$ meet at $(5,2)$, and find the value of $C$ there.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A potter makes $x$ mugs and $y$ bowls each day. The clay available gives $2x+3y\le14$ and the kiln time available gives $4x+y\le15$, with $x\ge0$ and $y\ge0$. The profit is $P=5x+4y$ pounds, which is to be maximised.

   (a)

   Show that the lines $2x+3y=14$ and $4x+y=15$ meet at $(3.1,\,2.6)$, and find the value of $P$ there.

   [3 marks]

   (b)

   The potter can only make whole mugs and bowls. Find the best production plan and the maximum profit.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A garden centre prepares $x$ trays of herbs and $y$ trays of flowers each day. Labour limits production to $2x+3y\le30$, greenhouse space gives $4x+2y\le30$, and seed supply gives $x\le6$, with $x\ge0$ and $y\ge0$. The profit is $P=6x+5y$ pounds, which is to be maximised.

   (a)

   Find the coordinates of the vertices of the feasible region, and hence find the maximum value of $P$ if $x$ and $y$ can take any non-negative values.

   [6 marks]

   (b)

   Trays cannot be split, so $x$ and $y$ must be integers. Explain why $(4,8)$ is not a solution, and find the optimal integer solution, explaining why no integer point does better.

   [6 marks]

   Total for question 4: 12 marks

End of questions