7 Linear programming —Part A

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Linear equations

Straight lines are the simplest kinds of graphs. These arise from linear equations.

The graph of a linear equation line can be drawn by calculating some values to ﬁnd the position of the line. We usually use three points to draw the line.

Linear equations

A linear equation has no powers. The graph of a linear equation is a straight line. The general form of a linear equation is

ax + by + c = 0 but it is often written as

y = mx + c

For instance, the equation 4x − 2y + 5 = 0 could be written as y = 2x + 21

2 ---.

!

Draw the graph of y = 2x − 3.

Solution

Make a small table with three well-separated x values. Calculate the values of y from the x values.

For x = −1, y= 2 × −1 − 3

= −5

For x = 2, y= 2 × 2 − 3

= 1

For x = 4, y= 2 × 4 − 3

= 5

Put the values in the table.

Example

1

y = 2x − 3

x −1 2 4

y

y = 2x − 3

x −1 2 4

y −5 1 5

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Equations of linear graphs are often given in other forms. For example, the equation may be given as 3x+ 2y= 24. We could use algebra to change this equation to y=−1.5x+ 12, but it may be easier to graph without changing it. We still substitute values to ﬁnd points, but we can choose either x or y values and work out the other. Values that give whole numbers will make it easier to plot the graph. If the variables are not x and y, as in 3m − 5n = 7, then we usually put the ﬁrst on the horizontal axis and the second on the vertical.

It is easiest to use graph paper to draw the graph. Draw the x- and y-axes long enough to use all the values shown in the table.

Label the axes and plot the points. Join the plotted points with a straight line. All three points should line up.

Write the equation of the line next to it.

6

5

4

3

2

1

−1

−2

−3

−4

−5

−2 1 2 3 4 5

y

x

−1

y = 2x − 3

Draw the graph of 3p + 2q = 24 by ﬁnding some points.

Solution

Treat p as horizontal and q as vertical.

Try p = 2. 3p + 2q= 24

3 × 2 + 2q= 24 6 + 2q= 24

2q= 24 − 6 2q= 18

q= 9

Write the point. (2, 9) is one of the points on the graph.

Try q = 6. 3p + 2 × 6= 24 3p + 12= 24 3p= 12

p= 4

Write the point. (4, 6) is on the graph.

Try q =−3 to get another point with whole numbers.

3p + 2 ×−3= 24 3p − 6= 24 3p= 30

p= 10

Write the point. (10,−3) is on the graph.

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When equations are given in the form shown in Example 2, we can use the intercept method to draw the graph. In this method, we just use the values x = 0 and y = 0 to ﬁnd where the line crosses the axes.

Write the points in a table.

Plot the points and draw the graph.

Use the correct variables on the axes, not x and y.

3p + 2q = 24

p 2 4 10

q 9 6 −3

10

8

6

4

2

−2

−4

4 6 8 10 12

q

p

2

3p + 2q = 24

0

Intercepts

The y-intercept is the point where a line crosses the y-axis (x = 0). The x-intercept is the point where a line crosses the x-axis (y = 0).

These points are called the intercepts. You may have to use fractions with this method.

!

Use the intercept method to draw the graph of 3x − 4y = 18.

Solution

Use x = 0 to ﬁnd the y-intercept. 3x − 4y= 18 3 × 0 − 4y= 18 0 − 4y= 18

−4y= 18

y= 18 ÷−4

y=−4

Write the intercept. (0, −4 ) is on the graph.

Use y = 0 to ﬁnd the x-intercept. 3x − 4y= 18 3x − 4 × 0= 18 3x − 0= 18 3x= 18

x= 18 ÷ 3

x= 6

Write the intercept. (6, 0) is on the graph.

1 2

---1 2

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The x-intercept is (6, 0) and the y-intercept is (0,−4 ). Plot these points on the axes and join to draw the graph. You can continue the graph past the axes.

Check using a third point.

It looks like (2,−3) is on the graph. Try it in the equation

3x − 4y= 18 Let x = 2 and y =−3.

3x − 4y= 3 × 2 − 4 ×−3 = 6 −−12

= 18, so the check is OK!

1 2

---2

−1

−2

−4

−5

−1 2 3 4 5 6

y

x

1

3x − 4y = 18

−3

7

Linear equations on a graphics calculator

Linear equations need to be in the form y = mx + c to draw them on a graphics calculator. The linear equation y = 2x − 3 may be graphed immediately, but the equation 3x − 4y = 18 may be changed as follows. Your teacher may show you other methods of changing the form of the equation, but all methods will give the same ﬁnal result.

3x − 4y= 18 Change so that the y is positive. 3x= 18 + 4y Isolate y. 3x − 18= 4y Reverse the equation. 4y= 3x − 18 Divide by 4 to get y. 4y ÷ 4= 3x ÷ 4 − 18 ÷ 4 Simplify. y= 0.75x − 4.5

All calculators

When entering a negative number such as −3, you must use the key. For example, you would use this in setting the view of the graph.

Casio CFX-9850GB PLUS

Choose the GRAPH menu and press .

Enter the equation Y = 2X − 3 by pressing 2

3 .

DRAW the equation by pressing .

(–)

EXE

X,θ,T

– EXE

y = 2x − 3 F6

Technology

_Graphics

calculator

Alternative

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Use Zoom ( ) to change the view of the

graph using IN ( ) and OUT ( ).

You can also use the V-Window ( ) to change the view of the graph so that −3 X 5 and −6 Y 6.

Press and DRAW ( ) to see the altered view.

Press and use the cursor keys to position the

highlight back onto Y1. Delete Y = 2X − 3 using DEL ( ) and YES ( ).

Now enter the equation Y = 0.75X − 4.5 by pressing 0.75

4.5 . Use the V-Window (

) to change the view to −2 X 8 and −6 Y 2.

DRAW the equation by pressing . Notice that the Y-intercept is −4.5.

Texas Instruments TI-83

Enter the equation Y = 2X − 3 by pressing 2

3 .

Draw the equation by pressing .

Use to change the view of the graph, using 2: Zoom In and 3:Zoom Out. You may need to move the

cursor before pressing .

You can also use to change the view of the graph so that −3 X 5 and −6 Y 6.

Press again to see the altered view.

QUIT by pressing . Delete Y = 2X − 3 by

pressing .

Now enter the equation Y = 0.75X − 4.5 by pressing

0.75 4.5 .

Use to change the view to −2 X 8 and

−6 Y 2. Draw the equation by pressing . Notice that the Y-intercept is −4.5.

Sharp EL-9650

The Sharp instructions are given on the CD-ROM.

SHIFT F2

F3 F4

SHIFT F3

y = 2x − 3

EXIT F6

y = 0.75x − 4.5

EXIT

F2 F1

X,θ,T – EXE SHIFT

F3

F6

Y= X,T,θ,n

– ENTER

y = 2x − 3

GRAPH

ZOOM

ENTER

WINDOW

GRAPH

2nd MODE

y = 2x − 3

Y= CLEAR

Y= X,T,θ, n – ENTER

WINDOW

y = 0.75x − 4.5

GRAPH

Calculator

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7.2

Construction of equations by

identifying variables and parameters

Graphs of linear equations can be used to solve simple problems by ﬁnding points on the line. In this case it may be necessary to translate the problem into a mathematical form by creating a mathematical model. Part of this process is to identify the important aspects of a real-life problem and express these in mathematical form. Some of these steps are

indicated below.

1 Draw graphs of the following linear equations using the table method.

a y = 3x − 2 b d = 2f + 3

c m = 6 − 2h d X =−k + 4

e C = 5 − 2x

2 Draw graphs of the following by ﬁnding some points.

a 2x + 4y = 10 b p − 2q = 6

c −x + 3y = 6 d 3m − 2v = 12

e 2a + c = 8

3 Use the intercept method to draw graphs of the following.

a 4x + 5y = 20 b p + q = 8

c 3c + 2d =−15 d 3d − 2e =−9

e 3y − x = 12

4 Use a graphics calculator to draw graphs of the following, and sketch the results.

a y = 3x + 4 b y =−2x + 3

c 5x + 2y = 10 d −3x + 4y = 12

e −2x + 4y = 7

Exercise 7.1

Linear equations

Additional exercise

7.1

Graphics

calculator

Using mathematical models to solve problems 1 Identify the quantities in the problem.

2 The quantities that change are called variables. State a letter name for each

variable—it is common to use the ﬁrst letter of the quantity unless this would cause confusion.

3 The quantities that do not change for the particular problem being considered are

called parameters. Different problems of the same type may have different parameters. It is often useful to identify parameters in order to solve similar problems more easily.

4 Write an equation (or several equations) that show the relationship(s) of the

variables and parameters.

5 Use the equations to solve the problem, perhaps using a graph as well.

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A soccer club plans to have a barbecue picnic. It costs $50 to book the site and buy gas for the big barbecue. The treasurer works out that it will cost about $8 per person for meat, rolls, soft drinks, etc. The club can afford to pay $300 for the barbecue.

a What are the variables? b List the parameters.

c Write a relationship for the cost of the barbecue.

d Draw a graph of the cost.

e Use the graph to ﬁnd the cost for 60 people.

f How many people could attend if the club paid the full cost?

Solution

a The variables are the number of people attending and the total cost.

b The parameters are the site and gas costs, the cost of food and drink for each person and

the amount the club can afford.

d We can use a table to draw the graph.

We know it must go to at least n = 60 because we want the cost for 60 people. Draw the graph.

Write letters for the variables. Let the number of people attending = n Let the cost of the barbecue = C

List the parameters. Site and gas cost = $50 Cost/head = $8

Maximum cost to the club = $300

c Write down the cost in words. The cost is the total of the site and gas cost and the cost of the food and drink. The cost of food and drink is $8 for each person.

Write using the variables and parameters. C = $50 + $8 × n

Simplify. C = 8n + 50

e Use n = 60 for the graph or equation. It would cost approximately $530 for 60 people.

f Use C = 300 on the graph. 31 people could attend if the club paid the full cost of $300.

n 0 40 80

C 50 370 690

700

600

500

400

300

200

100

0

20 40 60 80

Number of people (n)

T

otal cost of barbecue (

C

)

C = 8n + 50

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Exercise 7.2

Construction of equations

Modelling and problem solving

1 It costs a backpackers’ hostel $300 a night to make its 30 rooms available. The hostel

charges $40 a night for each room, but by the time it pays for cleaning and laundry it makes only $25 for each room occupied.

a Write the variables and parameters. b Write an equation showing the proﬁt. c Draw a graph showing the proﬁt.

d How much will the hostel make if 25 rooms are occupied? e How many rooms must be occupied to make $200?

f What is the break-even point—that is, the number of rooms that must be occupied to cover costs?

2 A remote weather station has a thermometer based on a resistance wire. At 25°C the wire

has a resistance of 10 Ω(ohms). The temperature must rise by 2°C for the resistance to increase by 1 Ω.

a Write the variables and parameters.

b Write an equation giving the temperature in terms of the resistance. c Draw a graph showing the temperature.

d What is the temperature when the resistance is 19 Ω?

e What is the temperature when the resistance is 35 Ω?

f What is the resistance for a temperature of 40°C?

3 A concrete pump can be hired

for $500 plus $200 a day.

a Write the variables and

parameters.

b Write the cost in terms of

the number of days hired.

c How much would it cost to hire the pump for 3 days?

d A pool construction

company has budgeted $1500 for the hire of a concrete pump. How many days’ use will this be?

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---7.3

Linear inequalities

Graphs of inequalities have to show a range of possibilities. This is done by shading one side of the graph of a linear equation. Graphs involving the symbols and include the values on the line, while graphs involving the symbols and do not include the values on the line. The graph of a linear inequality is drawn using the graph of the corresponding linear equation.

4 The water level in the big dam on a large property is at 6.5 m. The current rate of use means

that it drops by 0.2 m for every week without rain. The full amount the dam can hold is at a level of 8 m.

b Write an equation for the level of the dam in terms of the number of weeks without rain. c Draw a graph showing the level of the dam.

d What will the level be if there is no rain for the next 12 weeks? e After the last rain, the dam was full. When was the last rain?

f How long will it be before the dam is empty?

5 A mining company has two copper mines. The Agora mine produces ore containing 2%

copper and the Finbury mine produces ore with 3.5% copper. The company wants to produce 20 tonnes of copper each week.

b Write an equation showing the amount of ore from each mine that can be used. c Draw a graph of the amounts of ores used.

d How much Agora ore must be used if the maximum amount of ore from the Finbury

mine is 400 tonnes a week?

6 Cinema tickets cost $12 for adults and $8 for concessions. The total amount taken for one

session was $2400.

b Write the total takings for the session as an equation involving the number of adult

tickets and number of concession tickets

c Draw a graph of the equation.

d There were 150 concession ticketholders at the session. How many adult tickets were

sold?

7 A supplier of industrial gases charges a minimum fee of $80 for a service call, plus $20

for each gas cylinder that is reﬁlled. The supplier has a contract with a welding workshop for $500 worth of work each month.

b Write an equation for the monthly number of service calls and reﬁlls. c Draw a graph of the service calls and reﬁlls.

d One month the supplier makes 4 service calls to the workshop. How many reﬁlls can

the workshop have but still stay within the contract?

Linear inequalities

Linear inequalities are relationships involving greater or less instead of equality.

They are like equations but have the ‘=’ sign replaced by one of , , or .

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Showing linear inequalities

1 Replace the inequality sign by ‘=’ to make a linear equation.

2 Find the points for the linear equation.

3 Draw the linear equation as a solid or dashed line. For and , show the inclusion by a solid line. For and , use a dashed line to show that the values on the line are not included.

4 For ‘y ’ or ‘y ’, shade above the line. For ‘y ’ or ‘y ’, shade below the line. For inequalities expressed in other ways, choose a test point to ﬁnd the correct side to shade.

!

Draw the graph of y 3x − 2.

Solution

Replace by = and make a table.

Draw the graph with a dashed line for and shade above the line.

y = 3x − 2

x −1 2 4

y −5 4 10

10

8

6

4

2

−2

−6

−2 1 2 3 4 5

y

x

−1

−4

12

y > 3x − 2

Example

5

Draw the graph of 3x − 2y 12.

Solution

Replace by = to draw the line. 3x − 2y = 12

Find the intercepts. For x = 0, 3 × 0 − 2y = 12

−2y = 12

y =−6 For y = 0, 3x − 2 × 0 = 12 3x = 12

x = 4

The intercepts are (4, 0) and (0,−6).

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Since the inequality is , draw a solid line. Use (2, 2) and (4,−2) on either side of the line to ﬁnd which side should be shaded.

For (2, 2), 3x − 2y= 3 × 2 − 2 × 2 = 2

So 3x − 2y 12 is not true. For (4,−2), 3x − 2y= 3 × 4 − 2 ×−2

= 16 So 3x − 2y 12 is true. The side with (4,−2) should be shaded.

2

1

−1

−2

−3

−4

1 2 3 4 5

y

x

−1

−5

−6

3x − 2y 12

Linear inequations on a graphics calculator

Linear inequations can be graphed very similarly to linear equations on a graphics calculator. However, a dashed line cannot be shown on the Texas Instruments or Sharp calculators. To draw the graph of y 5 − 4x, follow the instructions below.

Casio CFX-9850GB PLUS

Choose the GRAPH menu and press .

Choose the TYPE Y by pressing ( )

.

Enter the rest of the inequation Y 5 − 4X by pressing

5 4 .

Use the V-Window ( ) to change the view of the graph so that −2 X 4 and −4 Y 8.

Press and DRAW ( ) to see the graph.

Texas Instruments TI-83

Set the to −2 X 4 and −4 Y 8.

Enter the equation Y = 5 − 4X by pressing 5

4 .

Move the cursor across to the \ line before Y1.

Press repeatedly until the ‘shade above’ sign

( ) appears. (For you would choose .)

Draw the equation by pressing .

Sharp EL-9650

The Sharp instructions are given on the CD-ROM.

EXE

F3 F6

F3

– X,θ,T _EXE

SHIFT F3

EXIT F6

WINDOW

Y=

– X,T,θ, n ENTER

ENTER

GRAPH

Technology

Graphics

calculator

Calculator

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7.4

Construction of constraints

In real problems, inequalities are used to specify restrictions on the possible solutions to a problem.

Draw graphs of the following inequalities, making sure you can also use a graphics calculator to draw some.

1 y 3x + 2 2 3x − y 9 3 4x + 3y −24

4 y 2x − 6 5 x + y −4 6 y 4 − x

7 x + 2y 10 8 y 3 − 4x 9 y − 5x 10

10 6x − 3y 15

Exercise 7.3

Linear inequalities

Additional exercise

7.3

Graphics

calculator

Constraints

A constraint is an inequality that restricts the possible solutions of a problem.

!

John has $20 and is buying ice-creams and chocolates for his children. Ice-creams cost $1.50 each and chocolates cost $2 each. Choose appropriate variables and write a constraint for the number of ice-creams that John can buy.

Solution

Write the variables. Let the number of ice-creams = i Let the number of chocolates = c

Write the total cost. Total cost = $1.50 × i + $2 × c

Use the $20 parameter. Total cost $20

Write the constraint. 1.5i + 2c 20

Example

7

Modelling and problem solving

1 Angelique owns a furniture factory. The factory has production runs making one item at

a time. It takes 10 minutes to make a chair and 25 minutes to make a table. The factory runs two shifts a day so there is a total of 15 hours of production time. Choose appropriate variables and write a constraint for the possible production of chairs and tables in a day. (Hint: Write the total time in minutes.)

2 Chan is buying meat for a club barbecue. He has been authorised to spend up to $200 on

hamburger patties and barbecue steaks. Hamburger patties cost $0.80 each and steaks cost $1.50 each. Choose appropriate variables and write a constraint to show the possible combinations of hamburgers and steaks he can buy.

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7.5

Regions defined by constraints

In most cases there is more than one constraint applying to a particular problem. In this case, the overlapping areas of the constraints will deﬁne the region of the graph that shows possible solutions.

3 Susan has been working at home preparing material for home delivery. It takes her 20 s

to place a magazine in a waterproof wrapper and 35 s to interleave advertising brochures. Choose appropriate variables and write a constraint to show how many of each she can do in half an hour.

4 Janine is buying a speedboat and trailer for use on the weekends. She can afford $6000 for

the trailer and boat together. Choose appropriate variables and write a constraint for the possible combinations of the costs of the boat and trailer.

5 Antoine needs to make at least $400 from selling his CDs and cassettes at a garage sale.

He has 30 CDs and 40 cassettes for sale. Choose appropriate variables and write a constraint for the combinations of prices of CDs and cassettes that he could use.

Mohammed can afford to spend up to $2000 on some new equipment for a sound studio. Microphones are $80 each and professional-quality headphones are $120 each. He must buy at least 12 microphones.

a Identify the variables and parameters for this problem. b Write constraints.

c Draw a graph and shade the region deﬁned by the constraints.

d Write down some possible combinations of microphones and headphones that

Mohammed can buy.

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Solution

Draw the graph with both lines solid. Shade both constraints, showing the overlap as the region deﬁned by the constraints.

a Write the variables. Let number of microphones = m

Let number of headphones = h

Write the parameters. Cost of a microphone = $80 Cost of headphones = $120

Maximum amount available = $2000

b Write the total cost. Total cost = $80 × m + $120 × h

Use the $2000 parameter. 80m + 120h 2000

Use the minimum microphone condition. m 12

Write the answer. The constraints are 80m + 120h 2000 and m 12, where m and h are respectively the number of microphones and

headphones bought.

c Treat m as horizontal and h as vertical. Find the intercepts for 80m + 120h = 2000.

80 × 0 + 120h= 2000 120h= 2000

h= 16 80m + 120 × 0= 2000

80m= 2000

m= 25

The intercepts of 80m + 120h = 2000 are (0, 16 ) and (25, 0).

Use (18, 8) to check which side to shade. For (18, 8), 80m + 120h= 80 × 18 + 8 × 120

= 2400 Since 2400 2000, (18, 8) is not in the shaded part, so the graph should be shaded

under the line.

Describe the second constraint. m 12 is shaded on the right of the vertical line through m = 12.

d Use the region of overlap. He can buy between 12 and 25 microphones and up to 8 headphones. The more microphones he buys, the fewer headphones he can afford.

2 3

---2 3

---16

14

12

10

8

6

4

2

2 4 6 8 10 12 14 16 18 20 22 24 26 28 18

m 12

80m + 120h 2000

Sound equipment

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Modelling and problem solving

1 Ursula has bought 8 sets of dining chairs from auctions of deceased estates and intends

to re-upholster and sell them. High-quality materials cost $300 for each set of chairs and medium-quality materials cost $200 for each set. Ursula intends to spend a maximum of $2000 on upholstery. She intends to upholster at least 2 sets in high-quality materials. Ursula may leave some sets to be upholstered later, after she has sold some.

a Identify the variables and parameters for this problem. b Write constraints.

d What are the possible combinations of upholstery?

2 Gavin is a professional potter and is making large and small mugs for sale at a craft fair.

He takes 10 minutes to make a small mug and 15 minutes to make a large mug. He has a maximum of 4 hours before he has to ﬁre the kiln to make sure the mugs are ready for Saturday. Gavin intends to make at least 5 large mugs.

d What are the possible combinations of mugs he can make?

3 Kam took his 4 children to Southbank for an afternoon. When he left home he had $30 in

his wallet. During the afternoon he bought 3 ice-creams and 2 chocolates for each of the children. He had 2 ice-creams and a chocolate himself. When he got home he was too tired to count how much he had left in his wallet. He knew that the chocolates cost at least $1.50 each.

d What are the possible costs of ice-creams and chocolates?

4 A ferry company is considering the purchase of some new ferries for an island run. Big

ferries will carry 120 passengers on each run and smaller ones will carry 80 passengers. Each run takes 3 hours and there are at least 600 passengers each day. There are 3 runs each day and the company is going to buy at least 2 small ferries. There have never been more than 1200 passengers on one day.

d Write some possible combinations of ferries that the company could purchase.

5 Juliette runs a mail-order business selling automotive tool kits. She advertises complete

kits for $60 each and emergency kits for $25 each. Over a period of time, Juliette ﬁnds that her sales vary between $900 a week and $1500 a week. She normally sells at least 10 complete kits each week.

d What are the possible combinations for the numbers of kits sold?

Exercise 7.5

Regions defined by constraints

Chapter

summary

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Chapter

Review

Communication and justification

1 What is meant by a linear equation?

2 What are the intercepts of a linear equation?

3 Explain what is meant by a variable.

4 What is a parameter for a problem?

5 What is the difference between a solid line and a dashed line on the graph of a

linear inequality?

6 Explain the meaning of the term ‘constraint’.

7 What is meant by the region deﬁned by the constraints of a problem?

Knowledge and procedures

8 Draw graphs of the following linear equations using the table method.

a y = 4x − 5 b v = 7 + 2f c z = 3 − 2t

9 Draw graphs of the following by ﬁnding some points.

a 3x + 5y − 18 = 0 b 2f − 3g + 9 = 0 c 5x − 2y =−7

10 Draw graphs of the following equations using the intercept method.

a 3x + 5y = 20 b d − 4c = 12 c 3g − 2h = 9

11 Use a graphics calculator to draw graphs of the following, and sketch the results.

a y = 3x + 5 b y = 3 − 4x c y = 3x + 1

12 Draw graphs of the following inequalities.

a 3x − 4y 24 b 5x + 2y 12 c y 3x − 6

Modelling and problem solving

13 Freda is running a donut stand at a local fair. She has to pay $300 for site rental and

it costs her 5 cents to make each donut. She sells the donuts for 30 cents each.

a Write the variables and parameters. b Write an equation showing the proﬁt. c Draw a graph of the equation.

d How many donuts does she need to sell to make an overall proﬁt of $200?

14 Keith has a backhoe and hires it out with a driver (himself) for $200 non-refundable

deposit plus $50 an hour.

a Write the variables and parameters. b Write an equation showing the cost of hire. c Draw a graph showing the cost of hire.

d Don budgeted $500 for trenches for a new building. For how long could he hire

the backhoe?

Ex 7.1

Ex 7.2

Ex 7.3

Ex 7.4

Ex 7.5

Ex 7.1

Ex 7.3

Ex 7.2

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15 A circus had gross takings of $5400 one night. It charged $10 for children and $16

for adults.

b Write an equation involving the numbers of adults and children attending. c Draw a graph of the equation.

d If 50 adults attended on that night, how many children were there?

16 Jose is doing the catering for his archery club’s annual children’s party. He has a budget

of up to $50 to spend on balloons and streamers. Balloons cost $2 a packet and streamers cost $3 a packet. Choose appropriate variables and write a constraint showing the possible purchases of balloons and streamers for the party.

17 Johanna is upgrading her ofﬁce equipment. New photocopiers cost $2500 each and new

workstations cost $3500 each. She can spend up to $12 000 but must buy at least 2 new photocopiers.

d What are the possible combinations of photocopiers and workstations?

18 Pauline is a collector of old movie serials. She has the opportunity to buy episodes of

cowboy serials at $200 an episode and episodes of old thrillers at $400 an episode. She can afford to spend $2000 and particularly wants 2 of the thriller episodes to complete a set.

d What are the possible combinations of purchases?

19 John is an outworker for a clothing company. He specialises in knitting Fair Isle pattern

jumpers but also does plain jumpers. It takes him 5 hours to do a Fair Isle jumper and 3 hours to do a plain one. He usually works a 60-hour week. The company he works for does not want more than 5 plain jumpers a week but will take any quantity of Fair Isle jumpers.

d What combinations of Fair Isle and plain jumpers is John able to make? Ex 7.2

Ex 7.4

Ex 7.5

7 Linear programming —Part A

Contents

Linear programming

Linear equations

Linear equations

Solution

Solution

Intercepts

Solution

Linear equations on a graphics calculator

Casio CFX-9850GB PLUS

Technology

Texas Instruments TI-83

Sharp EL-9650

SHIFT F2

X,θ,T – EXE SHIFT

– ENTER

2nd MODE

Y= CLEAR

Y= X,T,θ, n – ENTER

WINDOW

1 Draw graphs of the following linear equations using the table method.

Exercise 7.1

Using mathematical models to solve problems 1 Identify the quantities in the problem.

4 Write an equation (or several equations) that show the relationship(s) of the

d Draw a graph of the cost.

Solution

a The variables are the number of people attending and the total cost.

Maximum cost to the club = $300

Exercise 7.2

1 It costs a backpackers’ hostel $300 a night to make its 30 rooms available. The hostel

2 A remote weather station has a thermometer based on a resistance wire. At 25°C the wire

3 A concrete pump can be hired

Linear inequalities

4 The water level in the big dam on a large property is at 6.5 m. The current rate of use means

5 A mining company has two copper mines. The Agora mine produces ore containing 2%

6 Cinema tickets cost $12 for adults and $8 for concessions. The total amount taken for one

7 A supplier of industrial gases charges a minimum fee of $80 for a service call, plus $20

Linear inequalities

Solution

y =−6 For y = 0, 3x − 2 × 0 = 12 3x = 12

Casio CFX-9850GB PLUS

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Exercise 7.3

Constraints

Solution

1 Angelique owns a furniture factory. The factory has production runs making one item at

Regions defined by constraints

3 Susan has been working at home preparing material for home delivery. It takes her 20 s

a Identify the variables and parameters for this problem. b Write constraints.

Solution

a Write the variables. Let number of microphones = m

headphones bought.

d Use the region of overlap. He can buy between 12 and 25 microphones and up to 8 headphones. The more microphones he buys, the fewer headphones he can afford.

Modelling and problem solving

1 Ursula has bought 8 sets of dining chairs from auctions of deceased estates and intends

2 Gavin is a professional potter and is making large and small mugs for sale at a craft fair.

3 Kam took his 4 children to Southbank for an afternoon. When he left home he had $30 in

4 A ferry company is considering the purchase of some new ferries for an island run. Big

5 Juliette runs a mail-order business selling automotive tool kits. She advertises complete

Exercise 7.5

1 What is meant by a linear equation?

Knowledge and procedures

8 Draw graphs of the following linear equations using the table method.

Modelling and problem solving

13 Freda is running a donut stand at a local fair. She has to pay $300 for site rental and

d Don budgeted $500 for trenches for a new building. For how long could he hire

16 Jose is doing the catering for his archery club’s annual children’s party. He has a budget

d What are the possible combinations of photocopiers and workstations?

d What are the possible combinations of purchases?

d What combinations of Fair Isle and plain jumpers is John able to make? Ex 7.2