Formulating and graphical solution of LPsAQA A-Level Further Maths: Subtopic test

1. 1
   A workshop makes $x$ tables and $y$ chairs each week. Each table needs 4 hours of carpentry and 2 hours of finishing. Each chair needs 2 hours of carpentry and 3 hours of finishing. There are 40 hours of carpentry and 30 hours of finishing available each week. The workshop must make at least twice as many chairs as tables. The profit is £50 per table and £30 per chair, and the workshop wants to maximise its weekly profit.

   (a)

   Which inequality models the carpentry limit?

   [1 mark]

   • A$2x+4y\leq40$
   • B$4x+2y\geq40$
   • C$4x+2y\leq30$
   • D$4x+2y\leq40$

   (b)

   Which inequality models the rule on the number of chairs and tables?

   [1 mark]

   • A$x\geq2y$
   • B$y\leq2x$
   • C$y\geq2x$
   • D$y\geq x+2$

   (c)

   Write down the objective function and explain why $x\geq0$ and $y\geq0$ must be included as constraints.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A farmer mixes $x$ kg of feed $X$ and $y$ kg of feed $Y$ for each day. Each kilogram of $X$ has 3 units of protein and 1 unit of fibre and costs £3. Each kilogram of $Y$ has 2 units of protein and 2 units of fibre and costs £4. Each day the mixture must contain at least 24 units of protein and at least 12 units of fibre, and it must weigh at most 10 kg. The farmer wants to minimise the daily cost.

   (a)

   Which inequality models the fibre requirement?

   [1 mark]

   • A$2x+y\geq12$
   • B$x+2y\geq12$
   • C$x+2y\leq12$
   • D$3x+2y\geq12$

   (b)

   The feasible region is a triangle with vertices $(6,3)$, $(4,6)$ and $(8,2)$. What is the minimum daily cost?

   [1 mark]

   • A£30
   • B£33
   • C£32
   • D£36

   (c)

   Explain why a mixture of 5 kg of $X$ and 4 kg of $Y$ is not allowed.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Maximise $P=3x+2y$ subject to $2x+y\leq14$, $x+2y\leq10$, $x\geq0$ and $y\geq0$.

   (a)

   Find the coordinates of the vertices of the feasible region.

   [3 marks]

   (b)

   Use the objective line method to find the optimal solution and the maximum value of $P$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A school shop makes $x$ sandwiches and $y$ wraps each morning. A sandwich takes 3 minutes to prepare and a wrap takes 2 minutes, and there are 120 minutes of preparation time. At most 50 items can be made altogether, and at least 15 wraps must be made for the staff. The profit is £4 per sandwich and £3 per wrap.

   (a)

   Formulate this as a linear programming problem and use a graphical method to find the maximum profit.

   [6 marks]

   (b)

   The manager decides that at least as many sandwiches as wraps must be made. Show that the optimal plan in (a) changes, and find the new optimal plan and profit.

   [6 marks]

   Total for question 4: 12 marks

End of questions