Graphical solution of linear programsEdexcel A-Level Further Maths: Subtopic test

1. 1
   A cleaning company makes $x$ litres of cleaner A and $y$ litres of cleaner B each hour. The constraints are $2x+y\le12$ and $x+2y\le10$, with $x\ge0$ and $y\ge0$. The profit in pounds is $P=5x+4y$. The quantities do not need to be whole numbers.

   (a)

   Find the coordinates of the point where the lines $2x+y=12$ and $x+2y=10$ meet.

   [1 mark]

   • A$(4,4)$
   • B$\left(\frac{14}{3},\frac{8}{3}\right)$
   • C$(5,2)$
   • D$(6,0)$

   (b)

   What is the gradient of the objective line $5x+4y=k$?

   [1 mark]

   • A$\frac54$
   • B$-\frac45$
   • C$\frac45$
   • D$-\frac54$

   (c)

   Find the maximum profit.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A zoo buys $x$ kg of feed X and $y$ kg of feed Y each week. The requirements are $x+y\ge8$ (total mass) and $2x+5y\ge30$ (protein units), with $x\ge0$ and $y\ge0$. The weekly cost in pounds is $C=3x+4y$, which the zoo wishes to minimise. Unless stated otherwise, feed can be bought in any quantity.

   (a)

   Which of these points lies in the feasible region?

   [1 mark]

   • A$(6,4)$
   • B$(5,2)$
   • C$(4,4)$
   • D$(9,0)$

   (b)

   What is the minimum weekly cost, to the nearest penny?

   [1 mark]

   • A£32.00
   • B£45.00
   • C£28.67
   • D£29.00

   (c)

   The zoo can now only buy whole numbers of kilograms of each feed. Find the minimum cost and the quantities that give it.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A joiner makes $x$ shelves and $y$ stools each week. The wood constraint is $4x+3y\le26$ and the time constraint is $x+2y\le10$, with $x\ge0$ and $y\ge0$. Each shelf gives £3 profit and each stool gives £2 profit, so the weekly profit is $P=3x+2y$ pounds.

   (a)

   Use the vertex method to find the maximum profit if shelves and stools did not need to be whole numbers.

   [3 marks]

   (b)

   The joiner can only make whole numbers of shelves and stools. Find the optimal whole-number plan, explaining why simply rounding $(6.5,0)$ does not work.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A charity hires $x$ coaches and $y$ minibuses for a trip. A coach holds 50 people and a minibus holds 20, and at least 240 people must be taken. No more than 10 vehicles are available. A coach costs £300 and a minibus costs £100 to hire. The charity wishes to minimise the total hire cost $C$ (in pounds).

   (a)

   (i) Write down the two constraints (other than non-negativity), simplifying the capacity constraint so that it has no common factor.
   (ii) Find the coordinates of the vertices of the feasible region.

   [6 marks]

   (b)

   (i) Use the vertex method to find the minimum cost if $x$ and $y$ need not be whole numbers.
   (ii) The charity can only hire whole vehicles. Find the optimal whole-number hire and its cost, showing that the vertex rounded to $(1,9)$ is not suitable.

   [6 marks]

   Total for question 4: 12 marks

End of questions