D1: Linear programmingEdexcel International A Level Maths: Topic test

1. 1
   A print shop makes $x$ posters and $y$ leaflets each day. A poster needs $4$ minutes of printing and a leaflet needs $1$ minute. At most $240$ minutes of printing are available each day. The shop must make at least twice as many leaflets as posters.

   (a)

   Which inequality represents the printing time?

   [1 mark]

   • A$x+4y\le240$
   • B$4x+y\le240$
   • C$4x+y\ge240$
   • D$4(x+y)\le240$

   (b)

   Which inequality represents the number of leaflets compared with posters?

   [1 mark]

   • A$x\ge2y$
   • B$y\le2x$
   • C$y\ge2x$
   • D$y\ge x+2$

   (c)

   The profit is $30$ pence on each poster and $8$ pence on each leaflet. Write down the objective function, and state the conditions on $x$ and $y$ that arise because posters and leaflets are counted.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A florist makes $x$ bouquets and $y$ wreaths each day. Preparation time gives $x+y\le8$ and wire supply gives $2x+y\le12$, with $x\ge0$ and $y\ge0$. The florist wants to maximise the profit $P=3x+2y$, where $P$ is in tens of pounds.

   (a)

   Which of these points is a vertex of the feasible region?

   [1 mark]

   • A$(8,0)$
   • B$(0,12)$
   • C$(3,5)$
   • D$(4,4)$

   (b)

   What is the maximum value of $P$?

   [1 mark]

   • A$20$
   • B$18$
   • C$24$
   • D$16$

   (c)

   Find the coordinates of the point where the lines $x+y=8$ and $2x+y=12$ intersect.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A tailor makes $x$ shirts and $y$ jackets each week. A shirt needs $2$ hours of cutting and $3$ hours of sewing. A jacket needs $3$ hours of cutting and $2$ hours of sewing. There are at most $24$ hours of cutting and $25$ hours of sewing available each week. The profit is £$7$ on each shirt and £$6$ on each jacket. The tailor wants to maximise the weekly profit.

   (a)

   Formulate this as a linear programming problem, stating the objective function and all the constraints.

   [3 marks]

   (b)

   Ignoring the requirement that $x$ and $y$ are integers, the optimal solution is $x=5.4$, $y=4.4$ with $P=64.2$. Find the optimal integer solution, showing that no integer point does better.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A school hires $x$ minibuses and $y$ coaches for a trip. A minibus seats $15$ students and costs £$80$. A coach seats $50$ students and costs £$220$. At least $190$ students must be carried. At most $8$ vehicles can be hired altogether, and at least $2$ minibuses must be hired. The school wants to minimise the total cost $C$.

   (a)

   Formulate this as a linear programming problem. State the objective function and every constraint, including those that arise because vehicles are counted.

   [6 marks]

   (b)

   The feasible region, ignoring integers, has vertices $(2,3.2)$, $(2,6)$ and $(6,2)$. Find the minimum cost ignoring integers. Then explain why rounding that solution up to $(2,4)$ does not give the cheapest hire, and find the optimal integer solution.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A solar installer fits $x$ small systems and $y$ large systems each month. Crew time gives $x+2y\le20$ and panel supply gives $3x+y\le30$. At least two large systems must be fitted each month, so $y\ge2$, and $x\ge0$. The installer wants to maximise the profit $P=5x+4y$, where $P$ is in hundreds of pounds.

   (a)

   Which of these points is in the feasible region?

   [1 mark]

   • A$(6,5)$
   • B$(9,4)$
   • C$(4,9)$
   • D$(5,1)$

   (b)

   What is the maximum value of $P$?

   [1 mark]

   • A$54\frac23$
   • B$40$
   • C$72$
   • D$64$

   (c)

   The installer can fit only whole systems. Explain why the optimal solution of the problem, ignoring integers, is also the optimal integer solution.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A cycling club buys $x$ energy bars and $y$ gels for a race. A bar contains $40$ g of carbohydrate and a gel contains $25$ g. The club needs at least $1000$ g of carbohydrate and can carry at most $30$ items. A bar costs £$1.20$ and a gel costs £$0.90$. The club wants to minimise the cost.

   (a)

   Which inequality represents the carbohydrate requirement?

   [1 mark]

   • A$25x+40y\ge1000$
   • B$40x+25y\le1000$
   • C$40x+25y\ge1000$
   • D$40x+25y\ge30$

   (b)

   Which is the correct objective function, with $C$ in pence?

   [1 mark]

   • AMaximise $C=120x+90y$
   • BMinimise $C=120x+90y$
   • CMinimise $C=90x+120y$
   • DMinimise $C=x+y$

   (c)

   Show that buying $18$ bars and $12$ gels satisfies both constraints, and find its cost.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A school shop buys $x$ notebooks and $y$ calculators. The cost, in pounds, is $C=3x+4y$ and the shop wants to minimise it. The constraints are $x+y\ge8$, $x+3y\ge12$, $x\ge0$ and $y\ge0$.

   (a)

   Find the coordinates of the vertices of the feasible region.

   [3 marks]

   (b)

   (i) Find the minimum value of $C$ and where it occurs. (ii) The shop is now told it can buy at most $1$ calculator, so $y\le1$. Find the new minimum value of $C$ and where it occurs.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A boatyard builds $x$ dinghies and $y$ kayaks each week. The constraints are $2x+y\le14$ (assembly hours), $x+2y\le16$ (finishing hours), $x\le6$ (moulds), with $x\ge0$ and $y\ge0$. The profit is $P=4x+5y$, where $P$ is in hundreds of pounds, and the boatyard wants to maximise it.

   (a)

   Find the coordinates of every vertex of the feasible region and hence find the maximum value of $P$.

   [6 marks]

   (b)

   Because of a staff shortage the boatyard can build at most $9$ boats in total each week, so $x+y\le9$. Show that the previous optimal point is no longer feasible, and find the new optimal solution and the new maximum value of $P$.

   [6 marks]

   Total for question 8: 12 marks

End of questions