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C H A P T E R

18MODULE 3

Linear programming

How do we graph linear inequations?

How do we graph systems of linear inequations?

How do we transfer from an optimisation problem to its mathematical formulation?

How do we use graphical methods to solve linear programming problems with two

decision variables?

Linear programming is used to solve problems in a wide variety of areas, such as planning,

design and finance. There are now many computer programs available for solving linear

programming problems, and these are used extensively in industry. Linear programming is

often used to determine a solution to a problem to maximise profit or minimise cost. Such

problems are called optimisation problems.

18.1 Regions defined by an inequalityIn Chapter 16, lines in the cartesian plane were discussed. The lines are described by

equations, for example:

0 x = 1x

y = 2

y

x0 10

10x + y = 10

y

Each line is an infinite set of points. The position of each point can be described by

coordinates. For example, points with the coordinates (0, 2) (1, 2) (1.5, 2) and (√

2, 2) all lie on

the line with the equation y = 2. Points with the coordinates (2, 8) (3, 7) (0, 10) (3.6, 6.4) all

lie on the line with the equation x + y = 10.

488

SAMPLE

Cambridge University Press • Uncorrected Sample pages • 978-0-521-61328-6 • 2008 © Jones, Evans, Lipson TI-Nspire & Casio ClassPad material in collaboration with Brown and McMenamin


Chapter 18 — Linear programming 489

InequalitiesEach point in the shaded region has an

x coordinate greater than 2. The shaded

region is described as the region for which

x ≥ 2 and formally {(x, y) : x ≥ 2}. The

boundary (the line with the equation x = 2)

is included. The convention is that if the

boundary is included it is represented with a

continuous line and if it is not included it is represented with a dotted line.

x0

(4, 5)

(3, 2)(3, 1)

(2, 0)

x = 2x > 2

required regionboundary included

y

Example 1 Sketching regions with horizontal or vertical lines as boundaries

Represent each of the following regions with a suitable graph.

a x < 2 b y ≤ −5 c y > 4

Solution

a

x0

x = 2

2

y

x < 2

b

x

0–5 y = –5

y

y < 5

c

x

4

0

y = 4

y

y > 4

The line x + y = 10 separates the plane into two regions.

Each point in the shaded region has coordinates whose sum

is greater than or equal to 10, for example:

(2, 11) 2 + 11 = 13 ≥ 10

(10, 2) 10 + 2 = 12 ≥ 10

(10, 0) 10 + 0 = 10 ≥ 10

x0 10

10

(10, 2)

(2, 11)

x + y > 10

y

In the diagram opposite, each point in the shaded

region has coordinates whose sum is less than 10,

for example:

x

(–1, 10)

(0, 0)

10(1, 8)

10x + y = 10x + y < 10

y

−1 + 10 < 10

1 + 8 < 10

0 + 0 < 10

The region to be shaded can always be established

by testing one point. The origin is often the easiest point

to check.

SAMPLE



490 Essential Further Mathematics – Module 3 Graphs and relations

Example 2 Sketching regions defined by an inequality

Sketch the graph of the region 3x − 2y ≤ 6.

Solution

1 Sketch the graph of

3x − 2y = 6. The

straight line is drawn

as a continuous line

as the boundary

is included.

When x = 0, −2y = 6

∴ y = −3

When y = 0, 3x = 6

∴ x = 2 x(0, 0)

(2, 0)

(0, –3)

3x – 2y = 6

y

2 Check the origin (0, 0).

x0 2

–3

3x – 2y = 6

y

3 × 0 − 2 × 0 = 0 ≤ 6

(0, 0) is in the required region.

∴ the region above the line is

required.

3 Shade the required region.

Exercise 18A

1 Calculate the value of 5x + 6y corresponding to each of the given points and state whether

they are:

A in the region 5x + 6y > 22 B in the region 5x + 6y < 22

C on the line with equation 5x + 6y = 22

a (2, 2) b (3, 1) c (4, 2) d (0, 0)

2 Sketch the graph of each of the following regions:

a x ≥ 5 b x < 4 c x ≤ −2 d y > 4 e x > −5 f y ≤ 6

3 Sketch the graph of each of the following regions:

a y ≥ 2x + 5 b 3x − 2y < 6 c x + y ≤ 10 d x − y ≥ 10

e 2x − 6y ≤ 3 f y ≤ 2x g x − y > −3 h x − 2y > −3

i 2x + y ≥ 12 j 3x − 4y < 12

18.2 Regions defined by two inequalitiesExample 3 Sketching a region defined by two inequalities

Sketch the graph of the region that is defined by the inequalities x − y ≤ 10 and y ≤ x .

SAMPLE




Solution

1 Sketch the graph of the region

x − y ≤ 10.

2 On the same set of axes sketch

the graph of y ≤ x .

x

x – y = 10

10

–10

y

0

The region required is the region

with overlap of the two shadings.

x

y

x – y = 10

10

–10

0

y = x

required region

(boundaries included)

Example 4

Sketch the graph of the region that is defined by the inequalities 3x + 2y ≤ 6 and y − x ≤ 1.

Solution

x

y

3x + 2y = 6

y = x + 1

3

required region

(boundaries included)

5

4

5

9,

0 2

1 Draw the two straight lines.

2 Find the point of intersection of the

two lines by solving the simultaneous

equations or using a graphics calculator.

3 Shade each of the required regions.

4 The defined region is shown by the overlap.

Exercise 18B1 Sketch the region defined by each of the following sets of inequalities and find the

coordinates of the point of intersection:

a y ≥ −2x + 10

y ≥ x − 2

b 3x − 2y < 6

x − 3y > 9

c 2x + 3y > −6

3x − y < 6

d 2x + 3y ≤ 6

x ≥ 0

e 2y − 3x < 6

x < 0

f y − 3x < 6

x − y > −3

g x − y < 1

y − x < 1

h 2x − 2 ≥ y

2x ≤ 3 − 2y

i 2y < 4x + 2

y < 2x + 1

SAMPLE




18.3 Feasible regions

Example 5 Shading a feasible region

A hardware shop has room for no more than a total of 100 cans of paint of Brand X and

Brand Y. It is known that at least 30 cans of Brand X and no more than 50 cans of Brand Y will

be required.

By shading a suitable region of the cartesian plane, indicate the possible number of Brand X

and Brand Y that can be stored.

Solution

y

(60, 10)10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

feasible region

x = 30

x + y = 100

(40, 50)

(50, 50)

(46, 20)

(80, 1)

y = 50

(30, 50)

x

Let x be the number of cans of brand X.

Let y be the number of cans of brand Y.

The constraints can be expressed by the

following inequalities:

x ≥ 30

0 ≤ y ≤ 50

x + y ≤ 100

The shaded region in Example 5 is called the feasible region. It is the graph of all possible

pairs (x, y) which satisfy the inequalities.

Points with coordinates such as (30, 50) (40, 50) (46, 20) (50, 50) (60, 10) and (80, 1) all lie

within the feasible region. Points on the boundary are included in the feasible region.

Example 6 Shading a feasible region

Indicate, by shading, the region that satisfies all the following constraints. Mark in the

coordinates of the ‘corner’ points of the feasible region:

3x + 8y ≥ 24

3x + 2y ≥ 12

x ≥ 0

y ≥ 0

Solution

x

y

feasible region

0 2

2

4

4

6

6

(8, 0)

(0, 6)

3x + 2y = 123x + 8y = 24

8

83

, 2

10

SAMPLE




Exercise 18C

1 For each of the following indicate, by shading, the region that satisfies all of the constraints.

Mark in the coordinates of the ‘corner’ points of the feasible region.

a x + y ≤ 7

5x − 3y ≤ 15

x ≥ 0

y ≥ 0

b −x + 2y ≤ 12

y ≥ 2x − 9

x ≥ 2

y ≥ 0

c 3x + 4y ≤ 60

y ≥ 15 − 3x

x ≥ 0

y ≥ 3

d x + 2y ≥ 10

y ≤ 2x

0 ≤ x ≤ 6

0 ≤ y ≤ 14

e −4x + 5y ≤ 20

2x + 3y ≥ 6

x ≥ 0

y ≥ 0

f x ≤ 7

y ≥ 4

x + y ≤ 9

24x + 30y ≥ 360

g x + y ≥ 6

2x + 5y > 20

x ≤ 4

y < 5

h x ≥ 0

y ≥ 0

3x + y ≥ 15

2x + 3y ≥ 36

i x ≥ 0

y ≥ 0

x + y ≤ 4

x ≤ y

y ≤ 3

18.4 Objective functionsAn objective function is used to describe a quantity that you are trying to make as large as

possible (for example, profits) or as small as possible (for example, the amount of paint needed

to paint a house) subject to some constraints. Consider the following situation.

A hardware shop has room for no more than a total of 100 cans of paint of Brand X and

Brand Y. It is known that at least 30 cans of Brand X and no more than 50 cans of Brand Y will

be required. The profit on each can of Brand X is $2.00 and $2.50 for each can of Brand Y. If

the entire stock can be sold, how many of each brand should there be to yield maximum profit?

Let x be the number of cans of brand X.

Let y be the number of cans of brand Y.

The constraints for this situation can be expressed by the following inequalities:

x ≥ 30 (number of cans of Brand X required)

0 ≤ y ≤ 50 (number of cans of Brand Y required)

x + y ≤ 100 (total number of Paint required)

The profit function is given by:

P = 2x + 2.5y

where the value of P yields the number of dollars profit.

To be a solution, (x, y) must be in the feasible region.

Consider the point (60, 10).

For x = 60, y = 10 P = 2 × 60 + 2.5 × 10 = 145

SAMPLE




There are other values of x and y that yield a profit of $145

e.g. x = 55, y = 14; x = 50, y = 18; x = 45, y = 22; x = 30, y = 34.

All these points lie on the straight line 2x + 2.5y = 145 (a profit of $145).

For a profit of $185 the straight line is 2x + 2.5y = 185.

For a profit of $225 the straight line is 2x + 2.5y = 225.

All these lines are parallel. They have been added to the graph of the feasible region.

y

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

required region

x

2x + 2.5y = 2502x + 2.5y = 185

2x + 2.5y = 2252x + 2.5y = 145

(50, 50)

(30, 50)

It can be seen that the maximum profit possible is $225 and this is only achievable at one

point in the feasible region, i.e. (50, 50).

It can be seen that a larger value of P (e.g. P = 250) will not yield points in the feasible

region.

The function with the rule P = 2x + 2.5y is an example of an objective function. In linear

programming problems, the aim is to find the maximum or minimum value of an objective

function for a given feasible region. To help us do this, we can make use of the corner point

principle.

The corner point principleIn linear programming problems, the maximum or minimum value of a linear objective

function will occur at one of the corners of the feasible region.Note: If two corners give the same maximum or minimum value, then all points along a line joining the twopoints will also the same maximum or minimum values.

This means that we only need to evaluate the objective function at each of the corner points to

find the maximum or minimum value of an objective function.SAMPLE




Example 7 Finding the minimum value of an objective function

Find the minimum value of the objective function

Z = 3x + 2y subject to the constraints:

x ≥ 0, y ≥ 0

4x + y ≥ 12

3x + 15y ≥ 44

3x + 4y ≥ 22

as displayed in the feasible region opposite.

y

x0

feasibleregion

(0, 12)

(2, 4)143

, 2443

, 0

A

B

C

D

Solution

1 Write down the objective function.

2 Evaluate the objective function at each

of the corners.

3 Identify the corner point giving the minimum

value and write down your answer.

Z = 3x + 2y

A (0, 12) : Z = 3 × 0 + 2 × 12 = 24

B (2, 4) : Z = 3 × 2 + 2 × 4 = 14

C

(14

3, 2

): Z = 3 × 14

3+ 2 × 2 = 18

D

(44

3, 0

): Z = 3 × 44

3+ 2 × 0 = 44

The minimum value is Z = 14 when

x = 2 and y = 4.

Example 8 Finding integer solutions

A courier has to transport 900 parcels using large vans, which can take 150 parcels, and

smaller vans, which can take 60 parcels. The costs of each journey are $5.00 by large van and

$4.00 by small van. The total cost must be less than $44.

a Represent the feasible region for these constraints on a graph.

b Indicate the possible values of x and y on the diagram by using a ×.

c What is:

i the largest number of vehicles that could be used?

ii the smallest number of vehicles that could be used?

Solution

y

x0

5

10

15

2 4 6 8 10

150x + 60y = 900

5x + 4y = 44

boundary

excluded

(3.2, 7)

a Let x be the number of large vans and

y the number of small vans.

constraints:

x ≥ 0

y ≥ 0

150x + 60y ≥ 900 (5x + 2y ≥ 30)

5x + 4y < 44

SAMPLE




b The crosses in the feasible region indicate the possible numbers of small vans and large

vans.

c i The largest number of vehicles that could be used is 9: either 5 large and 4 small or 4 large

and 5 small or 6 large and 3 small or 7 large and 2 small.

ii The smallest number of vehicles that could be used is 6 large vans.

Example 9 Setting up and solving a linear programming problem

A business produces imitation antique vases in two styles: Ming Dynasty vases and Geometric

Period Greek vases.

Each vase requires:

potters to make the vase

artists to decorate the vase

During one week the business employs 10 potters and 4 artists. Each employee works for a

total of 40 hours from Monday to Friday.

The time spent making each vase is shown in the table below.

Employee Ming Geometric

Potter 8 hours 4 hours

Artist 2 hours 2 hours

Let x represent the number of Ming Dynasty vases made in a week and y represent the number

of Geometric Period vases made in a week.

a The business has a permanent weekly order of 10 Ming vases and 20 Geometric vases.

These give the inequalities

x ≥ 10 and y ≥ 20

The inequality formed from the total time available to the potters is

8x + 4y ≤ 400

Write down the corresponding inequality formed from the total time available to the artists.

b Draw the graphs of the four inequalities in part a.

c Clearly show the feasible region on the graph in part b.

The profit is $50 on each Ming vase is $50 and $30 on each Geometric vase.

d All vases produced in a week are sold. Write down an expression in terms of x and y for the

total profit $A that the business will receive.

e Determine the number of each type of vase that should be produced in a week to result in

the maximum profit.SAMPLE




Solution

y

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80x

x + y < 80

(20, 60)

(10, 20)

(10, 70)

x = 10

y = 20

8x + 4y = 400

(40, 20)

a 2x + 2y ≤ 160 or, equivalently, x + y ≤ 80 b, cd P = 50x + 30y

e Evaluate at each of the vertices of the feasible region:

(10, 70) P = 50 × 10 + 30 × 70 = 2600

(40, 20) P = 50 × 40 + 30 × 20 = 2600

(10, 20) P = 50 × 10 + 30 × 20 = 1100

(20, 60) P = 50 × 20 + 30 × 60 = 2800

To maximise the profit, 20 Ming vases and 60 Geometric

vases should be produced.

Exercise 18D1 The region that satisfies all of the following constraints: 5x – 2y = 20

–x + 2y = 8

x

y

AD

B C0

2468

10(7, 7.5)

2 4 6 8–2–2–4–6–8

–10

–4–6–8

5x − 2y ≤ 20

−x + 2y ≤ 8

x ≥ 0

y ≥ 0

is as shown.

a Write down the values of the coordinates of A, B, C and D.

b Find the maximum value of z = x + 2y subject

to the set of constraints above.

2 The region that satisfies all the following constraints

is shown:

x

y

103 C

–6

–4

–2

2

4

0B

6

8A E

D

y = 2x – 6

y = 6

4x + 5y = 40

4x + 5y ≤ 40

y ≥ 2x − 6

x ≥ 0

0 ≤ y ≤ 6

a Find the coordinates of A, B, C, D and E.

b Find the maximum value of z = 2x + y subject

to the set of constraints above.

3 a Illustrate the region that satisfies all the following constraints:

x + 3y ≤ 17

5x + 3y ≥ 25

0 ≤ x ≤ 8

0 ≤ y ≤ 6

b Find the maximum value of z = x + 3y subject to the set of constraints in a.

SAMPLE




4 The region that satisfies the following constraints

is shown:

x

y

0 3

4x + y = 12

3x + 4y = 22

3x + 15y = 44C

B

A12

443

4415

112

223

D

4x + y ≥ 12

3x + 4y ≥ 22

3x + 15y ≥ 44

x ≥ 0

y ≥ 0

a Find the coordinates of points A, B, C and D.

b Find the minimum value of z = 3x + 2y

subject to the set of constraints above.

5 a Illustrate the region that satisfies all the following constraints:

4x + 5y ≥ 52

y ≥ 0.5x

y ≤ 1.8x

x ≥ 4

y ≥ 0

b Find the minimum value of z = 4x + 10y subject to the set of constraints in a.

6 The region that satisfies all the following constraints

is as shown:

04x + 3y = 39

CB

D

x + 7y = 16

3x + y = 8

y = x + 6

x

y

A

3x + y ≥ 8

y ≤ x + 6

4x + 3y ≤ 39

x + 7y ≥ 16

x ≥ 0

y ≥ 0

a Find the coordinates of points A, B, C and D.

b Find the minimum value of z = 7.5x + 2.5y subject to the set of constraints above.

Questions 7−12 provide practice in setting up constraints and forming the objective function

7 A manufacturer makes two types of skateboard, A and B. Each skateboard is cut out in the

pattern shop, constructed in the assembly shop and then sanded and painted in the finishing

shop. Each skateboard of type A requires 1.2 units of labour in the pattern shop, 2.0 units

of labour in the assembly shop and 1.3 units of labour in the finishing shop. Each

skateboard of type B requires 1.0 units, 1.4 units and 1.5 units in the pattern, assembly and

finishing shops respectively. The numbers of units of labour available per week in the

pattern, assembly and finishing shops are 180, 240 and 200 respectively.

The profit from the sale of each skateboard of type A is $35, and from each skateboard

of type B is $30. The manufacturer can sell all the skateboards he can make. The number of

skateboards of type A made per week is a, and the number of skateboards of type B made

per week is b.

SAMPLE




The constraints are:

Pattern Shop: 1.2a + b ≤ 180

Assembly shop: 2a + 1.4b ≤ 240

a Give the inequation which describes the constraints given for the finishing shop.

b Give the rule for the profit per week, $P in terms of a and b.

8 A pheasant farmer wishes to blend food for her birds from two commercially available

products, Wheato and Oatex. Each kilogram of Wheato costs $1.80 and contains 200 units

of vitamins and 150 units of protein. Each kilogram of Oatex costs $2.10 and contains

180 units of vitamins and 260 units of protein. Each pheasant requires 12 units of vitamins

and 16 units of protein per day and the farmer has 290 pheasants.

The farmer wishes to satisfy her birds’ dietary requirements for the least cost. If x is the

number of kilograms of Wheato that she buys per day and y is the number of kilograms of

Oatex that she buys per day, write down the constraints on x and y.

Minimum units of vitamins required = 12 × 290

= 3480

∴ 200x + 180y ≥ 3480

Minimum units of protein required = 16 × 290

= 4640

a Give the inequality derived from the information about minimum protein.

b Give the rule for $C, the cost of the food per day.

9 A self-employed computer serviceman has more work than he can manage to do in a single

8-hour day. He can work for up to 5 hours on tasks of type A, which pay him $60 per hour,

and he can work for up to 6 hours on tasks of type B, which pay him $70 per hour. He has

at hand 41 new components and knows that he will use them at a rate of four per hour on

tasks of type A and six per hour on tasks of type B.

The serviceman wishes to decide the best mix of tasks for the day. If he spends x hours

on tasks of type A and y hours on tasks of type B, write down the constraints on x and y.

Also write down the formula for $P, the amount of pay received, in terms of x and y.

10 An ice-cream manufacturer makes 2000 litres per day of just three flavours: strawberry,

chocolate and vanilla. The profits on one litre of strawberry, chocolate and vanilla

ice-cream are, respectively, 50 cents, 40 cents and 30 cents. Demand is such that the

manufacturer should never make more than 600 litres of vanilla ice-cream in a day and

should never make more strawberry ice-cream than chocolate ice-cream.

On a particular day, the supply of raw materials is such that a maximum of 800 litres of

chocolate ice-cream can be made.

a If x and y are the numbers of litres of strawberry and chocolate ice-cream made that day,

respectively, how many litres of vanilla ice-cream are made?

b Write down, in simplest form, the constraints on x and y, and also the rule for the profit,

$P, assuming that all ice-cream made that day is sold.

SAMPLE




11 A manufacturer makes two articles, X and Y, which require three different component

parts, A, B and C. Each X requires 12A, 8B and 10C components, and each Y requires 9A,

11B and 13C components. The profit to the manufacturer on each X is $84 and on each Y

is $72.

In the period just before Christmas the manufacturer wishes to decide how many of X

and how many of Y to make, in order to maximise profit. There are 280A, 260B and 320C

components available at that time. If x is the number of X produced and y is the number of

Y produced, write down the constraints on x and y. Also write down the rule for the profit,

$P, assuming that all articles are sold.

12 A small service station sells petrol and diesel. The costs to the proprietor are 35 cents for

each litre of petrol and 42 cents for each litre of diesel. She can store at most 20 000 litres

of petrol and 15 000 litres of diesel at any given time. She has found from experience that

the demand for diesel is never more than half as great as the demand for petrol.

The proprietor wishes to arrange for deliveries of fuel on the day before a holiday

weekend. At that time her storage tanks are each at 50% capacity and she can afford to

spend $5000 buying fuel. If x is the number of litres of petrol she buys and y is the number

of litres of diesel she buys, write down the constraints on x and y.

13 For a journey across Ellesmene

Island in the Arctic Circle, an

explorer wishes to travel as lightly

as possible. He can obtain supplies

of two types of lightweight food,

type X and type Y, in order to supply him with energy, protein and carbohydrate. The

details about them are shown in the table.

Energy per Protein per CarbohydrateFood serve (units) serve (units) per serve (units)

Type X 600 3.0 20

Type Y 400 3.5 35

The explorer estimates that his minimum daily requirements of energy, protein and

carbohydrate will be 2600 units, 19 units and 150 units respectively. Each serve of type X

food weighs 36 g and each serve of type Y food weighs 56 g.

a If x and y are the number of serves per day of type X and type Y foods respectively that

the explorer takes, four of the constraints on x and y are:

x ≥ 0

y ≥ 0

600x + 400y ≥ 2600

3x + 3.5y ≥ 19

Give the constraint determined by the amount of carbohydrate required.

b If W grams is the weight per day of food that the explorer takes, write down the rule for

W in terms of x and y.

c Use a graphical method to determine how many serves per day of each food type the

explorer should take in order to minimise weight while still satisfying his daily dietary

requirements.

SAMPLE




14 Sam wants to buy some CDs, which cost $30 each, and some books, which cost $20 each.

She can spend up to $300 and wants to spend at least twice as much on CDs as she spends

on books. In addition, she wants to buy at least two CDs and no more than three books.

a If x denotes the number of CDs and y denotes the number of books that Sam buys, two

of the constraints that x and y must satisfy are:

3x + 2y ≤ 30

3x ≥ 4y

Write down two further constraints.

b Indicate on a graph the region for which x and y satisfy the constraints of a.

c Find the maximum number of items that Sam can buy.

15 A computer importer has two depots, D1 and D2,

and two retail outlets, R1 and R2. At the beginning

of October the importer receives orders for Versanote

180 computers, which can be summarised as shown:

Number of computersOutlet ordered

R1 123

R2 86

The numbers of computers available can be

summarised as shown:Number of computers

Depot available

D1 160

D2 115

Transport costs per computer can be summarised

as shown:Transport costs per

Journey computer (dollars)

From D1 to R1 2.20

From D1 to R2 2.80

From D2 to R1 1.50

From D2 to R2 1.70

The importer wishes to decide the most

economical way to supply exactly the right

number of computers to each retail outlet.

Let x be the number of computers moved

from D1 to R1 and let y be the number of

computers moved from D1 to R2.

a Write down, in simplest form, the constraints on x and y.

b Write down, in simplest form, the formula for the transport cost, $C, in terms of x and y.

c Use a graphical method to find the most economical way to solve the problem.

16 A mining company is required to move 200 workers and 36 tonnes of equipment by air. It is

able to charter two aircraft: a Hawk, which can accommodate 20 workers and 6 tonnes of

equipment, and an Eagle, which can accommodate 40 workers and 4 tonnes of equipment.

Let x denote the number of trips made by the Hawk aircraft and y denote the number of

trips made by the Eagle aircraft.

a Write down the constraints that x and y must satisfy.

b Indicate on a graph the values of x and y that satisfy the constraints of a.

c If the respective costs of the Hawk and the Eagle aircraft are $3000 and $4000 per trip,

find the number of trips that should be made by each aircraft in order to minimise the

total cost, and find the minimum cost.

SAMPLE




17 A produce grower decides to buy fertiliser containing three nutrients A, B and C to spread

on her paddocks. The minimum needs are 160 units of A, 200 units of B and 80 units of C.

There are two popular brands of fertiliser on the market: Fast Grow, costing $4 a bag,

contains three units of A, five units of B and one unit of C. Easy Grow, costing $3 a bag

contains two units of each nutrient.

If the grower wishes to minimise her costs while still maintaining the nutrients required,

how many bags of each brand should she buy?

18 An outdoor clothing manufacturer has 520 metres of polarfleece material. The

manufacturer will use it to make jackets of two types, Polarbear and Polarfox, to sell to

retailers. For each jacket of either type, 2.0 metres of material is required. However, the

Polarbear is simpler in design, requiring 2.4 hours each in the production process while

each Polarfox requires 3.2 hours. There are 672 hours available.

From past experience of demand, the manufacturer has decided to make no more than

half as many Polarfox jackets as Polarbear jackets. If the profit on each Polarbear jacket is

$36 and the profit on each Polarfox jacket is $42, use a graphical method to find how many

of each type should be made in order to maximise profit. What is this maximum profit?

19 The army is required to airlift 450 people and 36 tonnes of baggage by helicopter. There

are 9 Redhawk helicopters and 6 Blackjet helicopters available. Each Redhawk can carry

45 passengers and 3 tonnes of baggage, while each Blackjet can carry 30 passengers and

4 tonnes of baggage. Running costs per hour are $1800 for each Redhawk and $1600 for

each Blackjet. If the army wishes to minimise the cost of the airlift per hour, use a

graphical method to find how many of each helicopter should be used.

20 At the end of summer a small surfboard maker has 150 litres of polyester resin and

proposes to use it up making two types of surfboard for the next season. He can make

full-length boards, which require 5 litres of polyester resin each, and boogie boards, which

require 3 litres of resin each. He has at hand 200 metres of glassfibre cloth, and knows he

will need 6 metres of it for each full-length board and 4 metres for each boogie board. The

surfboard maker can obtain polyurethane blanks from a nearby supplier, but has found that

the supplier has only enough blanks for 25 full-length boards and 20 boogie boards. He is

confident that, as long as he makes more than two-thirds as many boogie boards as

full-length boards, he can sell all the boards he makes at a profit of $180 on each

full-length board and $110 on each boogie board.

Use a graphical method to find how many of each type of board he should make in order

to use up as much of the resin and cloth as possible while maximising profit. (You will need

to consider ‘integer points’ near the corner points of the feasible region.) How much resin

or cloth, if any, is left over with this optimal production scheme?SAMPLE



Review


Key ideas and chapter summary

Procedure for sketching The procedure for representing an inequality in a plane is as

follows:� Sketch the boundary line. The line is dotted if the

boundary line is not included and continuous if the

boundary line is included.

� Test with a point to determine which side of the line to

shade.

Example: x + y < 10

For the point (0, 0) 0 + 0 < 0

∴ shade underneath the boundary line.

x

y

x + y = 1010

10

0

(0, 0)(1, 8)

(–1, 0)

an inequality

Procedure for sketching The procedure for representing a system of inequalities for the

plane is as follows:� Graph each of the inequalities using a different shading

for each one.

ande.g� The overlap determines the required region.

Example: The region of points which satisfy y ≤ x

and x − y ≤ 10

x

y

x – y = 10

10

–10

0

y = x requiredregion

the region defined by asystem of inequalities

Feasible region The feasible region is the set of points that can be considered for

the given inequalities. For example, for the set of inequalities:

y ≥ 0

2x − y ≤ 0

x + y ≤ 30

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The feasible region is as shown.

y

x

feasibleregion

(0, 0) (30, 0)

(0, 30)

(10, 20)

2x – y = 0

x + y = 30

Linear programming Linear programming involves maximising or minimising a

linear quantity subject to the constraints represented by a set of

linear inequalities. The constraints define the feasible region in

which the quantity is to be maximised or minimised.

Objective function The objective function is a linear expression representing the

quantity to be maximised or minimised in a linear programming

problem. For example, P = 3x + 5y could be an objective

function representing profit.

The corner point principle The corner point principle states that in linear programming

problems, the maximum or minimum value of a linear objective

function will occur at one of the corners of the feasible region, or

on line on the boundary of the feasible region joining two of the

corners.

Maximum or minimum The maximum (or minimum) value of the objective function is

found by evaluating its values at the vertices (or along

boundaries) of the feasible region. (See the corner point

principle.)

Skills check

Having completed this chapter you should be able to:

sketch the region defined by inequalities

determine the feasible region defined by several inequalities

find the maximum or minimum value of an objective functionSAMPLE



Review


Multiple-choice questions

1 In the graph shown, the shaded region with

boundary included is the feasible region for

a linear programming problem that involves an

objective function P = 2x + 3y. The maximum

value of P for this feasible region occurs at a point

with coordinates:

A (0, 40) B (16, 40) C (36, 40)

D (48, 20) E (60, 0)x

y

(36, 40)

(48, 20)

(60, 0)

(0, 40)

0

(16, 40)

2 In which of the following graphs does the shaded region represent the points (x, y)

such that 5x + 3y ≤ 30? (In each case the boundary is included.)

A

x

y

06

10

B

0

3

5x

y C

x

y

0 3

5

D

6

0 10x

y E

x

y

0

3

5

3 The shaded region of the graph is described by:

A y + 3x ≥ 0 B y ≥ x + 1 C 3y − x ≥ 0

D 3y + x ≥ 3 E 3x − y ≥ 3

x

y

0

1

3

4 For the graph shown, the shaded region (boundaries

included) is the feasible region for a linear programming

problem that involves an objective function T = 2x + 3y.

The maximum value of T for this region occurs at the

point with coordinates:

A (0, 60) B (0, 50) C (20, 40)

D (30, 0) E (10, 60)x

y

(10, 50)

(0, 60)

(20, 40)

(30, 0)0

5 The region LMNO is shown in the diagram. If (x, y)

is a point in the region LMNO (boundaries included),

then the greatest value of P = 2x + 3y is:

A 8 B 12 C 13 D 15 E 16

x

y

O

L

M

N

1

123456

2 3 4 5 6

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6 Which of the following sets of inequalities describes

the shaded region in the diagram?

A x, y ≥ 0 2x + y ≥ 2 y ≤ x x + y ≤ 6

B x, y ≥ 0 2x + y ≤ 2 y ≥ x x + y ≤ 6

C x, y ≥ 0 2x + y ≥ 2 y ≥ x x + y ≤ 6

D x, y ≥ 0 2x + y ≤ 2 y ≤ x x + y ≤ 6

E x, y ≥ 0 2x + y ≥ 2 y ≥ x x + y ≥ 6x

y

1 6

6

2

y = x

0

7 The shaded region is described by:

A x ≤ 3, y ≤ 3, x ≥ 0

B x + y ≥ 3, x ≥ y, x ≥ 0

C x + y ≥ 3, x ≤ y, x ≥ 0

D x + y ≤ 3, x ≥ y, x ≥ 0

E x + y ≤ 3, x ≤ y, x ≥ 0x

y

y = x

x + y = 3

8 For point (x, y) in the region LMNO (boundaries

included) as shown, the smallest value

of 3x − 4y + 25 is:

A 4 B 9 C 10 D 25 E 46

x

y

O

L (0, 4)

(7, 0)

5

5

10N

M (3, 6)

9 Which one of the following sets of inequations describes

the shaded region (boundaries included) on the graph

shown?

x

y

0

6

8

5 7

A x ≥ 0, y ≥ 0

5x + 8y = 40

7x + 6y = 42

B x ≥ 0, y ≥ 0

8x + 5y = 40

6x + 7y = 42

C x ≥ 0, y ≥ 0

5x + 8y ≤ 40

7x + 6y ≤ 42

D x ≥ 0, y ≥ 0

8x + 5y ≤ 40

6x + 7y ≤ 42

E x ≥ 0, y ≥ 0

8x + 5y ≥ 40

6x + 7y ≥ 42

10 In the graph shown, the shaded region, with boundary

included, is the feasible region for a linear programming

problem. The equation of the objective function is

P = 2x + y. The maximum value of P for this

feasible region occurs at the point with coordinates:

A (0, 30) B (40, 80) C (40, 90)

D (20, 60) E (40, 0) x

y

(0, 0)

(0, 30)

(0, 40)(20, 60)

(40, 90)

(40, 80)

(40, 0)

SAMPLE



Review


Extended-response questions

1 Adam opens a new business. He employs several people to produce carved models

of two famous mathematicians, Descartes and Pythagoras. Each model is made of

wood and Adam employs people to:� carve the models (wood carvers)� paint the models (painters).

During one particular week,

Adam employs five wood

carvers and two painters. Each

worker works for a total of

40 hours from Monday to Friday.

The time spent for the carving and painting of each model is shown in the table.

Trade Pythagoras model Descartes model

Wood carver 4 hours 2 hours

Painter 1 hour 1 hour

Let x represent the number of Pythagoras models made in a week and y represent the

number of Descartes models made in a week.

a What is the maximum number of hours that can be spent carving the models in the

week?

b If each of the wood carvers only carve Pythagoras models, how many models

would they carve altogether in the week?

c The business has a permanent weekly order for 10 Pythagoras models and 20

Descartes models. These facts give the inequalities:

x ≥ 10

y ≥ 20

The inequality formed from the total time available to the wood carvers is

4x + 2y ≤ 200

Write down the corresponding inequality formed from the total time available to

the painters.

d Draw the graphs of the four inequalities in part c.

e Clearly show the feasible region defined by the four inequalities. [VCAA 2000]

2 Adam pays each of his workers an

hourly wage as shown in the table.Trade Payment per hour

Wood carver $24

Painter $20The cost of raw materials for each model is $10.

The profit on the Pythagoras model is $30 and

the profit on the Descartes model is $10.

a The selling price of a Pythagoras model is $156. Calculate the selling price of a

Descartes model.

b All the models produced in the week are sold. Write down an expression in terms

of x and y for the total profit $A that Adam will receive.

(cont’d.)

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c Use the graph from Question 1 and the expression from Question 2b, to

determine the number of models of each type that should produced in the week

to result in the largest possible profit for Adam.

d Find the maximum profit that Adam could make in the week. [VCAA 2000]

3 A small canning company produces two types of canned tuna with additional chilli.

The Super Tuna with fried chilli and the Elite Tuna with dried chilli. A can of Super

Tuna requires 200 g of tuna and 30 g of chilli. A can of Elite Tuna requires 300 g of

tuna and 20 g of chilli. The company can produce 800 cans of tuna a day. There is

200 kg of tuna available to them every day and no limit on the chilli. Let x be the

number of cans of Elite Tuna produced in a day and let y the number of cans of

Super Tuna produced in a day. The inequalities produced by this information are:

x ≥ 0

y ≥ 0

x + y ≤ 800

0.3y + 0.2x ≤ 200

a Sketch the graphs of these inequalities.

b Find the coordinates of the vertices of the region defined by these inequalities.

c If the company makes $1.00 profit on the Elite and $0.80 profit on the Super,

write an expression for the daily profit in terms of x and y.

d How many cans of each type of tuna should be produced to maximise the profit?

4 The Victory Vineyard makes both a

red wine and a white wine. The

table summarises the production

costs and sale prices per bottle of

the white wine and the red wine.

Type of wine Production cost Sale price

White $7 $15

Red $10 $20

The following conditions apply to the production of wine at the Victory Vineyard.� The maximum total number of bottles of red wine and white wine

produced by the Victory Vineyard in any day is 700.� The total production cost of red and white wine cannot exceed $6400 per day.� The maximum number of bottles of red wine that can be produced is 570 per

day.

a Use R to represent the number of bottles of red wine produced each day and W to

represent the number of bottles of white wine produced each day. These

conditions are then expressed algebraically by:

Constraint 1 R + W ≤ 700

Constraint 2 10R + 7W ≤ 6400

Constraint 3 R ≤ 570

SAMPLE



Review


The lines bordering regions of the R-W plane defined by these constraints are

shown and labelled on the graph. Identify the line representing each constraint

equation.

R0 200 400 600

200

400

600

800

1000

A BC

Y

W

800

b Find the coordinates of the point Y.

c Given that the profit per bottle is $8 for white wine and $10 for red wine, write

down an expression for $P, the total profit for the sales of the wine, in terms of

W and R.

d Find the maximum daily profit that can be earned by the Victory Vineyard from

selling their wine. [VCAA 2001]

SAMPLE


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