Formulating linear programsEdexcel A-Level Further Maths: Subtopic test

1. 1
   A workshop makes $x$ tables and $y$ chairs each week. Each table needs 3 hours of cutting time and each chair needs 2 hours. There are at most 20 hours of cutting time available each week.

   (a)

   Which inequality represents the cutting-time constraint?

   [1 mark]

   • A$3x+2y\ge20$
   • B$2x+3y\le20$
   • C$3x+2y\le20$
   • D$3x+2y=20$

   (b)

   A slack variable $s_1$ is introduced so that the constraint becomes an equation. Which equation is correct?

   [1 mark]

   • A$3x+2y+s_1=20$
   • B$3x+2y-s_1=20$
   • C$3x+2y+s_1\le20$
   • D$3x+2y=20+s_1$

   (c)

   Find the value of $s_1$ when $x=4$ and $y=3$, and state what it represents in context.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A farmer mixes $x$ kg of feed A and $y$ kg of feed B. Each kilogram of either feed supplies 1 unit of a vitamin, and the mixture must supply at least 5 units of the vitamin in total.

   (a)

   Which inequality represents the vitamin requirement?

   [1 mark]

   • A$x+y\le5$
   • B$x+y\ge5$
   • C$x+y=5$
   • D$x-y\ge5$

   (b)

   A surplus variable $s_2$ is introduced to turn the constraint into an equation. Which equation is correct?

   [1 mark]

   • A$x+y+s_2=5$
   • B$x+y+s_2\ge5$
   • C$-x-y+s_2=5$
   • D$x+y-s_2=5$

   (c)

   An artificial variable $t_1$ is also added to this constraint. Write down the equation and explain why $t_1$ is needed.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A company makes $x$ standard gift boxes and $y$ deluxe gift boxes each day. A standard box needs 4 minutes of packing and 2 metres of ribbon; a deluxe box needs 6 minutes of packing and 5 metres of ribbon. Each day there are 240 minutes of packing time and 150 metres of ribbon available, and at least 10 boxes must be made. Profit is £3 per standard box and £5 per deluxe box. The company wishes to maximise its daily profit $P$ (in pounds).

   (a)

   Formulate this situation as a linear programming problem.

   [3 marks]

   (b)

   Use the simplified packing constraint $2x+3y\le120$. Write each constraint as an equation using slack, surplus and artificial variables as appropriate, and write the objective function in the form used in a Simplex tableau (ignore the artificial variable).

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A farmer plants $x$ hectares of wheat, $y$ hectares of barley and $z$ hectares of oats. There are 80 hectares available in total. Labour is limited to 200 hours: wheat needs 3 hours per hectare, barley 2 hours and oats 4 hours. A contract requires at least 20 hectares of wheat. Profit per hectare is £400 for wheat, £300 for barley and £350 for oats. The farmer wishes to maximise the total profit $P$ (in pounds).

   (a)

   (i) Formulate this as a linear programming problem.
   (ii) If every constraint is written as an equation, state how many slack, surplus and artificial variables are needed.

   [6 marks]

   (b)

   The farmer proposes planting $x=30$, $y=20$ and $z=10$.
   (i) Show that this plan is feasible by finding the values of the slack variables $s_1$ (land) and $s_2$ (labour) and the surplus variable $s_3$ (wheat).
   (ii) Find the profit.
   (iii) Interpret $s_1$, $s_2$ and $s_3$ in context.

   [6 marks]

   Total for question 4: 12 marks

End of questions