Formulating linear programsEdexcel International A Level Maths: Subtopic test

1. 1
   A bakery makes two types of bread. Each batch of loaf A needs 3 kg of flour and 2 hours of oven time. Each batch of loaf B needs 2 kg of flour and 4 hours of oven time. Each day there are 60 kg of flour and 80 hours of oven time available. The profit is £20 per batch of A and £30 per batch of B. The bakery makes $x$ batches of A and $y$ batches of B each day, and wants to maximise its daily profit.

   (a)

   Which inequality represents the flour available?

   [1 mark]

   • A$2x+3y\le60$
   • B$3x+2y\le60$
   • C$3x+2y\ge60$
   • D$3x+2y\le80$

   (b)

   Which inequality represents the oven time available?

   [1 mark]

   • A$4x+2y\le80$
   • B$x+2y\le80$
   • C$2x+4y\le80$
   • D$2x+4y\ge80$

   (c)

   Write down the objective function, saying whether it is to be maximised or minimised, and write down the non-negativity conditions.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A farmer mixes two feeds, X and Y, for cattle. Feed X costs £0.40 per kg and contains 3 units of protein and 1 unit of vitamin per kg. Feed Y costs £0.60 per kg and contains 1 unit of protein and 2 units of vitamin per kg. Each day the mixture must provide at least 15 units of protein and at least 12 units of vitamin, and the farmer can store at most 20 kg of mixture. The farmer uses $x$ kg of X and $y$ kg of Y each day, and wants to minimise the daily cost.

   (a)

   Which inequality represents the protein requirement?

   [1 mark]

   • A$3x+y\ge15$
   • B$3x+y\le15$
   • C$x+3y\ge15$
   • D$x+y\ge15$

   (b)

   Which is the objective function?

   [1 mark]

   • AMaximise $0.4x+0.6y$
   • BMinimise $0.6x+0.4y$
   • CMinimise $3x+y$
   • DMinimise $0.4x+0.6y$

   (c)

   Write down the inequality for the vitamin requirement and the inequality for the storage limit.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A workshop makes $x$ tables and $y$ chairs each week. A table needs 6 hours of carpentry and 1 hour of painting. A chair needs 2 hours of carpentry and 1 hour of painting. There are 120 hours of carpentry and 40 hours of painting available each week. The workshop must make at least 4 tables and at least three times as many chairs as tables. The profit is £50 per table and £15 per chair, and the workshop wants to maximise its weekly profit. Only whole tables and chairs can be made.

   (a)

   Write down the inequalities for the hours of carpentry and painting, and for the non-negativity of the variables.

   [3 marks]

   (b)

   Write down the inequalities for the two conditions on the numbers made, the objective function, and any other condition the variables must satisfy.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A charity packs two types of food parcel: standard and large. A standard parcel contains 2 kg of rice and 1 kg of beans, and feeds 4 people. A large parcel contains 3 kg of rice and 3 kg of beans, and feeds 7 people. The charity has at most 600 kg of rice and at most 450 kg of beans. At least 40 standard parcels must be packed, and the number of large parcels must not exceed the number of standard parcels. The charity wants to feed as many people as possible.

   (a)

   Formulate this as a linear programming problem, defining your variables and stating the objective, all the constraints, and any other conditions.

   [6 marks]

   (b)

   (i) Show that $x=120$, $y=80$ satisfies all the constraints, and find the number of people fed.
   (ii) The charity has a budget of £1400. A standard parcel costs £6 and a large parcel costs £10. Write down an inequality for the budget, and explain whether $x=120$, $y=80$ is still feasible.

   [6 marks]

   Total for question 4: 12 marks

End of questions