Linear programmingIB MYP Maths Extended: Subtopic test

1. 1
   A minibus carries $x$ adults and $y$ children. It can carry at most 14 people in total, there must be at least 2 adults, and the number of children can be at most three times the number of adults.

   (a)

   Which inequality shows that the minibus carries at most 14 people?

   [1 mark]

   • A$x+y\le14$
   • B$x+y\ge14$
   • C$x+y<14$
   • D$xy\le14$

   (b)

   Which inequality shows that the number of children is at most three times the number of adults?

   [1 mark]

   • A$x\le3y$
   • B$y\ge3x$
   • C$y\le x+3$
   • D$y\le3x$

   (c)

   Determine whether 5 adults and 9 children can travel in the minibus. Give a reason.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A workshop makes $x$ tables and $y$ chairs each week. The weekly limits are $2x+y\le16$ (wood) and $x+2y\le14$ (labour hours), with $x\ge0$ and $y\ge0$. The profit is $P=30x+20y$ dollars. The feasible region has vertices $(0,0)$, $(8,0)$ and $(0,7)$, plus one more vertex where the two boundary lines meet.

   (a)

   Find the coordinates of the vertex where the lines $2x+y=16$ and $x+2y=14$ meet.

   [1 mark]

   • A$(4,6)$
   • B$(6,4)$
   • C$(7,2)$
   • D$(5,6)$

   (b)

   What is the maximum weekly profit?

   [1 mark]

   • A$\$240$
   • B$\$180$
   • C$\$260$
   • D$\$380$

   (c)

   Show that the point $(8,7)$ is not in the feasible region.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A student investigates how the best choice changes when the objective function changes. The feasible region is defined by $x\ge0$, $y\ge0$, $x\le6$, $y\le5$ and $x+y\le8$. The objective is to maximise $P=x+ky$, where $k$ is a positive constant.

   (a)

   Find the coordinates of the two vertices of the feasible region that lie on the line $x+y=8$. Find the maximum value of $P$ when $k=2$, and the vertex where it occurs.

   [3 marks]

   (b)

   Find the vertex or vertices that give the maximum value of $P$, and that value, for $k=0.5$, $k=1$ and $k=3$. Describe the pattern and explain it.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A charity packs two types of food parcel. Each Type A parcel contains 4 kg of rice and 2 kg of lentils and feeds 5 people. Each Type B parcel contains 2 kg of rice and 3 kg of lentils and feeds 4 people. The charity has 40 kg of rice and 36 kg of lentils. It packs $x$ Type A parcels and $y$ Type B parcels, where $x\ge0$ and $y\ge0$, and wants to feed as many people as possible.

   (a)

   Write down two inequalities for the rice and lentil limits, find the vertices of the feasible region, and find the largest number of people that can be fed. Treat the parcels as whole numbers.
   (The inequalities $x\ge0$ and $y\ge0$ do not need to be repeated.)

   [6 marks]

   (b)

   A volunteer says: 'Type A parcels feed more people each, so we should pack only Type A.' Evaluate this claim, using your results from (a).

   [6 marks]

   Total for question 4: 12 marks

End of questions