1. science
2. mathematics

Uncorrected Sample Pages
NCEA Level 2
THETA
MATHEMATICS
DAVID BARTON
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The Theta Mathematics package
Foreword to students, parents and teachers
Investigations
Puzzles
CAS calculator applications
Simulations
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CONTENTS
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THINK
THETA
Co-ordinate geometry
Further co-ordinate geometry
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2.1 Co-ordinate geometry 3
4
5
6
7
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2.2 Graphical models
Polynomials and their graphs
Functions – domain and range
Other mathematical functions and their graphs
Transformations of graphs and the connection with parameters
Trigonometric graphs
Piecewise graphs
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ix
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1
2
15
34
35
49
62
91
104
134
2.4 Trigonometric relationships
142
9
10
11
12
143
159
170
186
Triangle trigonometry
The sine rule
The cosine rule
Circular measure
2.6 Algebraic methods 207
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209
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Basic algebra
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Algebraic fractions
Factorising and quadratic expressions
Solving equations and rearranging formulae
Quadratic equations
Exponential expressions
Logarithms 225
231
240
252
271
283
294
20
21
22
23
Introducing differentiation
Calculus and curves
Anti-differentiation Calculus applications 295
307
319
340
2.12 Probability methods 359
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Probability Further probability The normal distribution 360
378
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2.7 Calculus methods D
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2.13 Simulation
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438
Simulation methods 2.14 Systems of equations
459
28 29
460
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Simultaneous equations
Non-linear simultaneous equations A nswers
Index
Useful formulae
Normal distribution table
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The Theta Mathematics package
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2.1
Theta Mathematics has been completely updated to reflect the current requirements of Mathematics
and Statistics in the New Zealand Curriculum. It has been reorganised to provide full coverage of the
14 Level 2 Mathematics and Statistics Achievement Standards that, in 2012, replace the earlier nine.
Most students enter for only six or seven of these Achievement Standards in any given year. This
means that, effectively, they are only covering about half the curriculum in a year-long course and,
therefore, they can study topics in more depth. Their learning resources need to be comprehensive
and offer a wide spectrum of skills, challenges and interpretation-type problems. Moreover, schools
now have the flexibility to offer a partial Level 2 course to Year 11 students or to allow students, in
some cases, to gain Level 2 NCEA credits over two years. Therefore, for modern classrooms, the
­traditional ‘one size fits all’ textbook is not appropriate. Not only would most students use just half
of it, but a textbook that covered all 14 Achievement Standards would either be hopelessly overweight or, if its size were kept in bounds, would skim over topics without doing them full justice.
Our challenge is to provide a solution that meets the needs of students and schools, offering them
flexibility and not dictating a particular course. At the same time, all 14 Achievement Standards
have to be covered comprehensively and rigorously so that school communities can have confidence
that students are well prepared for assessment. We also have to bear in mind the growing number
of schools who expect to deliver the curriculum and learning resources digitally and want the
­opportunity to select the particular combination of modules from the Theta Mathematics package that
meets their needs.
The two-path solution we offer is aimed at several groups of students:
• students planning to do mathematics in Year 13, and therefore requiring a course that offers the
appropriate prerequisites for Level 3 Calculus and/or Level 3 Statistics
• students wanting to gain some NCEA Level 2 credits in Year 11
• students aiming for a full collection of Level 2 Mathematics credits spread over a two-year period.
The table shows how the 14 Achievement Standards fit into the two textbooks at the centre of
the package.
Theta Dimensions
2.1
2.2
2.4
2.6
2.7
2.12
2.13
2.14
2.2
2.3
2.4
2.5
2.8, 2.9, 2.10
2.11
2.12
2.13
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Theta Mathematics
Co-ordinate geometry
Graphical models
Trigonometric relationships
Algebraic methods
Calculus methods
Probability methods
Simulation
Systems of equations
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Graphical models
Sequences and series
Trigonometric relationships
Networks
Statistical investigation
Statistical report
Probability methods
Simulation
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The Theta Mathematics package
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Theta Mathematics:
•
•
•
•
•
includes the topics that lead on to more advanced mathematics in Year 13
includes the topics that are currently assessed externally
provides complete foundations for NCEA Level 3 Calculus
provides a thorough preparation for the more demanding parts of NCEA Level 3 Statistics
is the best choice for students who want to leave their options open so that they can take either or
both of Level 3 Calculus and Level 3 Statistics.
Theta Dimensions:
includes the few topics that tend to terminate at this level
offers a mathematically worthwhile course fully at NCEA Level 2 standard
provides all the prerequisites and background needed for NCEA Level 3 Statistics
is a good choice for students who may want to take statistics in Year 13.
Each textbook contains a broad selection of Achievement Standards – more than the six that most
students will enter – which means a particular course is not dictated by either textbook.
Each textbook has an accompanying workbook (Theta Mathematics Workbook and Theta Dimensions
Workbook). Furthermore, all four printed publications have a single, overarching, digital teaching
resource (Theta Mathematics Teaching Resource) that supports them all.
2.1
2.2
2.4
2.6
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2.7
2.12
Dedication
This book is dedicated to my brother,
John Campbell Barton.
2.13
In loving memory of John,
9 December 1952–21 June 2005
2.14
Who would true valour see, let him come hither
David Barton
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Foreword to students, parents and teachers
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Welcome to Year 12 and another year of learning mathematics.
This textbook will provide you with full coverage of the following NCEA Mathematics
Level 2 Achievement Standards: 2.1, 2.2, 2.4, 2.6, 2.7, 2.12, 2.13 and 2.14.
Features to particularly note in the treatment of these Achievement Standards include the
following:
• expanded coverage of geometric reasoning in Achievement Standard 2.1
• new material on functions, and domain and range in Achievement Standard 2.2
• new material on piecewise functions in Achievement Standard 2.2
• more emphasis on key features of graphs in Achievement Standard 2.2
• completing the square has been added in Achievement Standard 2.6
• graphs of derived functions are now covered in Achievement Standard 2.7
• a new section on relative risk in Achievement Standard 2.12
•full explanations of how to use appropriate technology to carry out normal probability
calculations in Achievement Standard 2.12, and normal distribution tables have been
included in an appendix to save teachers from having to provide separate books of tables.
As you can see by looking at the edge of this textbook, each Achievement Standard has its
own coloured section so that you know exactly which topics fit where!
Each section has full explanations, worked examples and plenty of exercises so that you
can learn new skills and solve mathematical problems, with a solid and comprehensive
core of graded, skill-based problems as in previous editions. Where relevant, the activities
are carefully integrated with application-style questions in contexts that relate to students’
experiences.
Updated features of this edition of Theta Mathematics include:
• liberal and functional use of colour
• new application questions added throughout
• full emphasis on using relevant technology
•new investigations and puzzles, as well as many of the popular ones from the previous
edition
•spreadsheets that can be downloaded from the companion website,
www.mathematics.co.nz
•tags on the edge of each page for easy navigation between exercise questions and answers
• references to CAS activities, where appropriate.
Year 12 is where students first begin to study mathematics in some depth. Until this
stage, much of mathematics has been applications-based, with some straightforward skills
work. Now, students start to learn about some of the underlying structure of mathematics
and study it for its own sake. At this level, mathematics is an optional subject, and we can
assume that most students using this textbook have enjoyed doing mathematics so far.
My aim in updating this textbook has been to continue to stimulate students’ interest and
encourage them to carry on with mathematical studies.
Topics that students are meeting for the first time, such as calculus and the normal
distribution, are accompanied by a large number of graded, skills-based questions intended to
reinforce students’ understanding and build solid foundations for future learning.
Full and comprehensive answers are provided. In most cases, for questions involving
numerical working, answers are correct to four significant figures so that students can check
whether their calculations are accurate. In some cases, however, answers have been rounded
appropriately to fit the context of the question.
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Foreword to students, parents and teachers
Although this textbook has been organised around the NCEA Level 2
Achievement Standards listed above, it also provides carefully planned
coverage of selected mathematics and statistics achievement objectives in the
New Zealand Curriculum. NCEA is about assessing the curriculum – but it is
also important to provide a course that lays solid foundations for future study
and builds on mathematics covered earlier. Theta Mathematics and its sister
textbook, Theta Dimensions, both follow a ‘spiral’ and integrated approach
to learning mathematics – revisiting topics frequently to reinforce earlier
understanding, and gradually increasing the difficulty level. The treatment of
topics in these textbooks addresses the fact that the strands in NCEA, although
assessed separately, are inter-related in many places.
ARTER
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INVESTIGATION
Advice to students
Mathematics is a subject you learn by doing. You will feel more confident about
it when you have attempted a wide range of problems. The exercises in this
textbook have been graded so that the easier ones come first. Check the answers
frequently as you go, to get feedback on whether you understand the process.
Use the tags on the edge of each page to help you navigate between the exercise
questions and the answers.
Write as much working as possible – this makes it easier for someone else to
check your understanding, and explain where you need help. Do not become
discouraged if you make careless mistakes sometimes – some of the most famous
mathematicians have been hopeless at telling the time or managing money!
Throughout the textbook, you will find a variety of investigations, puzzles
and spreadsheet activities. Try doing these – you will be surprised at how often
interesting mathematics is found in unexpected and unfamiliar situations.
You will probably use this textbook mainly in the classroom. However, you
need to study and do extra activities in your own time to do as well as possible.
The Theta Mathematics Workbook is ideal for this purpose. It provides full
homework and revision for the course, with plenty of NCEA-style questions.
The workbook is closely referenced to this textbook, making it really easy to
match homework and revision with what you are doing in class. Because the
workbook is a ‘write-on’ publication, you can add your own notes, highlight
important points, colour-code places where you made mistakes for future
reference or add hints from your teacher. The Theta Mathematics Workbook also
features a student CD, which includes key worked examples from the textbook,
all the spreadsheets referred to in this textbook, and links to internet websites.
Mathematics will lie behind much of your career and journey through life.
With a broad mathematics education and a positive attitude, you should do well.
Best wishes for an enjoyable, challenging and successful year.
David Barton
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STARTER
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PUZZLE
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TECH
TECHNOLOGY
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A
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CAS CALCULATOR
DATA
DATA SET
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BLACKLINE MASTER
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INVESTIGATIONS
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140
146
154
157
172
174
177
181
190
196
205
213
234
Hollow squares
Three squares in a row
Floor joists
Play and rewind
The Golden Ratio
Quadratic equation solver
Tartaglia and Cardano
The pendulum
The Towers of Hanoi
The shrinking chord
Combining trapeziums
The warehouse
The horse-breeder’s paddock
Fifty cents and the polar bears
Coffee cups
The normal curve and the witch of Agnesi
Off to the freezing works
The acoustic concert hall
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58
70
85
90
112
237
239
244
251
262
265
270
273
280
298
331
351
358
394
404
425
463
481
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The roller-coaster
Reading age
The dance floor
Exponential and logarithmic graphs
Different bases, different curves
Producing trig graphs on a spreadsheet
Modelling monthly temperatures in Clyde,
Central Otago
Room numerals
The Cairo pentagon
Square roots and trigonometry
What is the value of tan(90°)?
Pythagoras and the cosine rule The octagon chop
The Golden Ratio and the cosine rule
Maximising the apparent width
The Polygon Society and the round tablecloth
The Auckland–Sydney road tunnel
The Reuleaux triangle Chequered flags
Splitting rectangles
D
Wayward perpendicular bisectors
The stolen bicycle
Circumscribed triangle
Double triangles
The mobile crane
Straight hands
The pool ball triangle
The lune
The cucumber puzzle
The third highest number
The four friends
Stuffing envelopes
Xavier and Yvonne
The shoemaker’s will
The grandstand
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PUZZLES
29
33
153
179
185
190
193
203
205
216
218
222
224
227
233
Filling the bath
A windy day
The bicycle puncture
Pumpkin patch
Minding your ps and qs
The model railway
This century base 3
Fours galore
Car number plates
Last digit
Two and two make five
The white marbles
Two flat tyres – tow me home! The chocolate rabbits
Penny wise, pound foolish
The Boeing 757
The great picnic puzzle
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246
248
257
260
265
272
273
275
278
279
369
373
463
463
464
467
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ix
A
S
CAS CALCULATOR APPLICATIONS
Applications of linear rearrangements
Applications of index equations
Multiples of powers of x
Differentiating simple products and
quotients
Finding the point where the gradient
has a given value
The rule for anti-differentiation
of polynomials
Definite integrals
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290
300
301
308
334
397
411
416
429
468
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The licorice factory and the triangle
inequality
Ten to one it’s Pythagoras
The cars and the goat (the Monty
Hall problem)
Normally distributed values
Heights of children
Volumes of soft drinks
Excess baggage charges
Cats and mice (predator–prey)
Sharks and fish (predator–prey)
Ferrets and fantails (predator–prey) Off-road vehicles and toheroa
(predator–prey)
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Fifty cents and the polar bears
Tossing a fair coin
Waiting for heads
Fives and sixes
The petrol station and the
coloured cards
The World Series
Boys and girls
Three members of a family
The gamblers
The Chinese Lunar Calendar
The Zodiac
The dartboard
The overbooked car-ferry
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SIMULATIONS
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328
Applications of area
Using technology to find the
standard deviation
Using technology to calculate
standard normal probability
Using technology to calculate any
normal probability
Using technology to obtain normal
‘inverses’
Intersection of lines and curves
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24
2.1 Co-ordinate geometry
Intersection of lines
Some topics in mathematics blend algebra and geometry together. When working out where lines
intersect, you can use:
• geometry − draw the lines and inspect the graph to locate the point of intersection, or
• algebra − solve a pair of simultaneous equations, where each equation represents one of the lines.
The approach here uses methods that are also covered in
Achievement Standard 2.6: Equations and expressions.
Ideally, when locating the intersection of two lines, consider a combined
approach:
• drawing a graph to get an approximate idea of the location
• using algebra to determine the exact co-ordinates.
Pairs of lines do not always intersect on the grid (with integer values) and, in
some cases, only algebra will give you the exact values.
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Example
Determine the point of intersection of the two lines:
5x − 2y − 10 = 0
3x + y − 4 = 0.
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Answer
Use technology or otherwise to draw the graph of both lines on the same grid.
Note that the technology may require you to enter the equations of the lines in the form
of y = mx + c:
y = 5 x + 5
(1)
2
−
y = 3x + 4
(2)
The graph shows that the lines intersect near the point (1.5, −1). We can use the ‘Trace’ function on
a CAS calculator to obtain a very accurate estimate.
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56
2.2 Graphical models
Absolute value
The absolute value of a number can be thought of as its size or the distance it is from 0 on a number
line. We always take distance as being non-negative.
The absolute value function is written as f(x) = |x|.
• If x < 0, then |x| = −x, making it positive.
• If x ≥ 0, then |x| = x (so that it stays non-negative).
The effect of the absolute value brackets | … | is that they make the result of the expression inside
y
positive if it is negative.
Answers
a 3 Note that the domain, or set of possible x-values, for |x|
is the real numbers, R.
The range of f(x) = |x| is {y: y ≥ 0, y ∈ R}.
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The term ‘graphing software’ here is
very broad. It could include the use of
graph-drawing programs, a graphics
calculator, a spreadsheet or a CAS
calculator. The graphs can also be
drawn by hand.
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From a spreadsheet:
TECH
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Example
Use graphing software to draw
the graph of f(x) = |x + 3| − 5.
Identify the key features of the
graph, including the domain, the
range and any intercepts.
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The graph of the absolute value
function has only non-negative
y-values:
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Examples
Evaluate the following.
a |−3|
b |14|
c |10 − 17|
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Answer
On a CAS calculator:
C
A
S
Note that the
command for
absolute value
is ABS( ).
9781442549487_CH-4.indd 56
Note that the formula for the absolute
value of the number in any cell, e.g. B1,
is =ABS(B1).
The domain is all the real numbers, R,
because there are no restrictions on the
values of x.
The range is y ≥ −5.
The y-intercept is (0, −2).
The x-intercepts are (−8, 0) and (2, 0).
The vertex of this absolute value graph is
(−3, −5).
6/12/11 6:26 AM
139
8 Piecewise graphs
3 The outline of a symmetrical arch design can be modelled by two
parabolas. The design starts at a height of 6 m above the ground
and is 6 m wide at the bottom. The height, h, in metres of the
design above the ground can be modelled by a piecewise function,
where x is the distance in metres from the left edge.
 −1 x 2 + 2x + 6,

h(x) =  4
 −1 2
 4 x + px + q,

0≤x≤k
on
k≤x≤6
a What is the value of k?
bCalculate the height above ground level of the top of the
design.
c What are the values of p and q?
dExplain whether the two parts of the design meet at an angle or as part of a smooth curve.
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aCalculate the pressure at the time when the leak occurs.
bWhen t = 38 seconds, the pressure is 100 kPa. Use this information to write the equation of
the hyperbola (that is, find the values of a and b).
cWrite the equation of the horizontal asymptote of the hyperbola. Explain what this
asymptote represents in the context of this model.
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t
Time (seconds)
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Straight line:P = 11t for values of t between 0
and 20.
Hyperbola: P = a for values of t greater
t−b
than 20.
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P
Tyre pressure (kPa)
4 A tyre-fitter is testing a tyre to check whether
it is faulty. The tyre inflates normally and
then a leak occurs. The pressure (P) in kPa
(kilopascals) can be modelled by a straight line
and a hyperbola.
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5 Under ideal conditions, a rat population increases at 25% per month. After five
months, some bait is laid that contains a contraceptive (birth control) substance
and, one month after that, the population starts to decrease by 10% per month.
The curves and line in the diagram have the following equations:
2 y = p
3 y = k(0.9)x
for values of x < 5
for values of x between 5 and 6
for values of x > 6.
aUse equation 1 to calculate the maximum population
and hence, find the value of p.
bUse the graph to estimate the two times (to the nearest
month) when the population is 400 rats.
cExplain what you would expect to happen to the
population of rats eventually, assuming equation 3
continues to apply. In your answer, mention a property
of an exponential curve.
dCalculate the value of k.
9781442549487_CH-8.indd 139
2
700
Population
1 y = 200(1.25)x
600
500
3
1
400
300
200
100
0
2
4
6
8 10 12 14
Time (months)
6/13/11 7:54 AM
146
2.4 Trigonometric relationships
23 A and B are two points on the circumference
of a circle with centre O. C is a point on
OB such that
B
AC ^ OB.
A
5 cm
AC = 12 cm
12 cm C
and BC = 5 cm.
Calculate the
size of ∠AOB,
marked q on the
diagram.
24 A square with side length 5 cm is sitting
on top of a square with side length 12 cm,
as shown in
the diagram.
5 cm
What is the
area of the blue
quadrilateral?
12 cm
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INVESTIGATION
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The Cairo pentagon
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The diagram shows a tessellation made up of identical
equilateral pentagons. It is reputedly named after Cairo,
the capital of Egypt, because the streets there feature
similar tiling patterns.
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1 Explain why the Cairo pentagon is not regular.
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2 Two of the angles of the Cairo pentagon are 90°.
Calculate the sizes of the other three. Note: none of
the other three angles is 120°!
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Applications of right-angled triangles
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Right-angle trigonometry has many applications:
• navigation
• building
• surveying
• civil engineering.
Example
When roads go round bends, engineers design them with a ‘camber’. The camber makes it easier
for cars to stay on the road when cornering.
At one place on a curve, a road has a camber of 4° and is 6.6 m across. Calculate the difference
in height between the sides of the road.
Answer
There is enough information in the diagram to draw a
right-angled triangle. The difference in heights is marked x.
Use the sin formula.
9781442549487_CH-9.indd 146
o = h × sin(A)
x = 6.6 × sin(4°)
= 0.46 m (2 sf)
x
6.6 m
4°
6/13/11 8:57 AM
12 Circular measure
203
PUZZLE
The lune
A lune is a figure shaped like a crescent moon. In plane
geometry, a lune is a region bounded by two arcs.
Calculate the area of the lune shown in the diagram. The
centres of the two arcs that define the lune are at A and B.
B is the midpoint of CD.
×
D
B
6 cm
6 cm
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4 A convex lens is designed so that its crosssection is represented by the intersecting
part of two circles, each with a radius of
110 mm. The arcs on the edge of the lens
subtend angles of 49° at the centre of each
circle. Calculate the area of the cross-section,
to the nearest mm2.
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1 A stage at a theatre has the shape of a
segment with a centre angle of 3 radians
and a radius of 10 m. Calculate the area of
the stage.
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EXERCISE
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Applications of the area of a segment
Convex lens
49°
49°
110 mm
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2 The diagram here shows the cross-section of
a lens. The lens has been cut from a sphere
with a radius of 400 mm, and the angle at
the centre of the sphere subtending the arc is
0.21 radians. Calculate the area of the crosssection.
3 An equilateral triangle has sides of 10 cm.
A circle (circumcircle) passes through all
three vertices of the triangle.
a Calculate the radius of the circumcircle.
bCalculate the total area of all three
segments formed by the triangle and
circumcircle.
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5 Two circles overlap as shown below. The
radius of one circle is 50 cm, and the radius
of the other circle is 70 cm. The length of the
chord common to both is 30 cm. Calculate
(to 2 sf) the area that lies in both circles.
30 cm
6/13/11 1:06 AM
Calculate the volume of the prism.
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8 The photo shows a
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cylindrical fuel tank. The
S
A
diameter of the tank is 4 m.
When the volume of fuel
drops below 20% full, then more fuel should
be ordered. What is the depth of fuel in the
tank when it is 20% full? Give your answer
to the nearest cm.
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6 A road tunnel
has a crosssection that can
be modelled by
a major segment
3m
3m
of a circle with
2.3
radius 3 m.
Tunnel floor
The tunnel
floor subtends
an angle of 2.3 radians at the centre of the
circle. The tunnel is straight and is 68 m
long.
aCalculate the height of the top of the
tunnel above the tunnel floor.
bCalculate the width of the tunnel
floor.
cEstimate the volume of earth and
rock removed to construct the tunnel.
Give your answer to the nearest
hundred m3. Show your working and
explain what you are calculating at
each step.
dExplain why the estimate may not be
very accurate.
co
py
12
2.4 Trigonometric relationships
rig
204
D
ra
ft
on
ly
-
7 A prism has a cross-section made up of
an isosceles trapezium and a segment, as
shown in the diagram.
• The centre of the segment is at
the intersection of the diagonals
of the trapezium, and the radius
is 4.5 m.
• The isosceles trapezium has parallel
sides measuring 7.2 m and 4.8 m, with
a distance of 4.5 m between the parallel
sides.
• The depth of the prism is 3 m.
3m
7.2 m
4.5 m 4.5 m
4.5 m
3m
4.8 m
9781442549487_CH-12.indd 204
4m
Fuel
(tank is
20% full)
d
Note: to determine the required depth, you
will need either to use a CAS calculator that
solves equations involving the formula for
the area of a segment, or to use ‘guess and
check’ methods; and then use trigonometry.
9 Two circles
overlap so that
the centre of each
×O
one lies on the
O1 ×
2
circumference
of the other.
Each circle has
a radius of 10 cm. Calculate the area of the
intersection, shaded orange in the diagram.
6/13/11 1:07 AM
17 Quadratic equations
265
PUZZLE
The model railway
ar
s
on
32
m
A model railway runs on a circular track inside a square section of land. The railway
is adjacent to all four sides of the square and the closest it runs to any corner of the
square is 32 m. Determine the area of the square section of land. Give your answer to
the nearest m2.
INVESTIGATION
Pe
Quadratic equation solver
17
ly
-
co
py
rig
ht
Let’s create a spreadsheet that automatically solves quadratic equations (in the form
ax2 + bx + c = 0) when we type in the values of a, b and c into columns A, B and C.
We’ll use columns D and E for the two solutions.
The spreadsheet extract shows what the headings and part of the first row should
look like.
−b
+ b 2 − 4ac .
2a
ra
ft
x=
on
The formula for the first root is:
D
Using cell references, we enter a similar formula in cell D3:
=(-B3+SQRT(B3^2–4*A3*C3))/(2*A3)
1 Open a spreadsheet program and type in the
headings and the values 1, 4 and -7 (for the first
equation) in cells A3, B3 and C3.
2 Enter the formula in cell D3. If you don’t get the
value 1.31662479, check that you have typed the
formula correctly!
3 Write the formula for the second root. Enter it in cell
E3. What value do you get?
4 Enter the values of a, b and c for equations 2–10 in
the block of cells A4:C12.
5 Copy the formulae in cells D3 and E3 downwards
nine times.
Here are the 10 quadratic
equations we will solve
using the spreadsheet.
1 x2 + 4x - 7 = 0
2 3x2 + x - 8 = 0
3 8x2 - 24x - 23 = 0
4 x2 - 14x + 49 = 0
5 2x2 + 8x + 21 = 0
6 9x2 + 6x + 1 = 0
7
-6x2
8
x2
9
6x2
10
x2
-x-8=0
+ 5x - 6 = 0
- 5x - 3 = 0
+x+1=0
6 Print out the result.
9781442549487_CH-17.indd 265
6/11/11 7:51 PM
304
2.7 Calculus methods
Notice that:
• f ′(x) is 0 between 0 and 10, and also between 40 and 50 – this is where the track is level
• when the roller coaster is going up (height increasing), the gradient function f ′(x) is positive
• when the roller coaster is going down (height decreasing), the gradient function f ′(x) is
negative.
The best place to draw the derived function graph, or gradient graph, is immediately below the
function graph.
Example
on
y
f(x) = x2
x
Pe
ar
s
f ′(x) = 2x
rig
ht
x
co
py
The equation of the function need not be known to sketch the graph of the derived function. Just
examine the graph of the original f(x) function, and read off or estimate the gradient at various
x-values. Then transfer the value of the gradient to the corresponding x-value below.
Example
-
f(x)
ly
20
y
–2
f′(x)
ra
ft
2
y=x+2
4
D
y=1
T
I
P
9781442549487_CH-20.indd 304
on
y = x2 – 8x + 18
y=2
x
y = 2x – 8
x
Proceeding from left to right:
•the first part of f(x) has a gradient of 1 so,
below this part of the function, we draw the
graph of y = 1
•then, the horizontal portion of the graph of
f(x) clearly has gradient 0, so we draw the
line y = 0 below this part
•finally, the parabolic portion on the right
will have gradient 2x - 8, so we draw a line
with gradient 2
Note that the gradient is undefined when x = 0,
so the derived function graph has no value at
x = 0. The empty (or open) circles at x = 0 show
that 0 is excluded from the domain of f(x).
Take a look at some applets that draw the graph of the derived function for
various functions. Each applet shows how the gradient changes as you move
along a graph. These applets are linked to from the Theta Mathematics links
page at www.mathematics.co.nz.
DATA
6/13/11 1:31 AM
348
2.7 Calculus methods
9 A train passes a signal at a speed of 12 m/s.
The train’s acceleration t seconds after
passing the signal is (4 - 2t) m/s2, until it
comes to rest.
a For how many seconds after passing the
signal does the train continue to gain
speed?
b Find, in terms of t, an expression for the
speed, v m/s, of the train.
c What is the greatest speed attained by
the train after it has passed the signal?
d How long after passing the signal does
the train take to stop?
e How far past the signal did the train
travel?
down does the aircraft take to reach normal
taxiing speed?
ly
a If the speed of the spacecraft was 10 m/s
at the moment the engines were fired,
find an expression for the speed, v m/s,
of the spacecraft at time t.
b In terms of the spacecraft, what
2
information would be given by
v dt ?
D
ra
ft
on
23
-
co
py
rig
ht
Pe
ar
s
on
7 At a certain moment on its journey, the
rocket engines of a spacecraft were fired.
Its acceleration, a m/s2, t seconds after that
moment is given by a = 3t - 1.
∫0
8 A machine is driving piles for a wharf in a
river estuary. It does this by lifting a heavy
weight 4 m directly above the pile, and
then releasing the weight onto the pile. The
impact drives the pile into the riverbed. The
acceleration of the weight is 4.9 m s-2.
a What is the initial velocity of the
weight?
b How long does the weight take to reach
the pile? (Hint: find an expression
involving distance s, and then substitute
s = 4.)
c Find the velocity of the weight when it
hits the pile.
9781442549487_CH-23.indd 348
10 A boat is travelling at a constant speed
of 12 m s-1 towards a marina. The speed
limit in the marina is 2 m s-1 so the boat
decelerates when it passes a buoy. The
acceleration of the boat t seconds after
passing the buoy is given by the rule:
a=
−t
.
5
a How long does the boat take to show
down to the speed limit?
b Calculate the distance travelled by
the boat after passing the buoy until it
reaches the speed limit.
6/13/11 3:25 AM
384
2.12 Probability methods
7 This table gives information about simplified
weather forecasts over a 90-day period in a
region of New Zealand.
Calculate the probability that the weather
forecast was correct on any given day.
Forecast
Dry
Actual
weather
Dry
Wet
Total
Wet
Total
49
8
57
5
28
33
54
36
90
Risk and relative risk
The risk that a New Zealander will currently have a diagnosis of Type 2 diabetes (as
at July 2010) was about 4.1%.
That information is not particularly useful overall unless it is considered together
with risk factors. These risk factors are a mixture of lifestyle and other factors, and
sometimes they are linked and not independent.
py
ARTER
-
co
ST
rig
ht
Pe
ar
s
on
Risk can be thought of as probability. The term ‘risk’ is usually used in connection with events that
have unpleasant consequences (e.g. death, serious disease, etc.) and, although some of the events
may be subject to uncertainty, sometimes there are factors relating to exposure or otherwise that can
either reduce the risk or make it worse.
For example, the risk of injury or death from a serious road crash is reduced for someone who
wears a safety-belt, and is increased for those who drive after drinking alcohol or who use a cell
phone while driving. Whether a crash occurs or not is still uncertain (subject to some randomness or
unpredictability) but some of the factors associated with driving can either increase or decrease the
probability that the crash occurs.
on
Overweight
ly
At increased risk
Normal weight
Older than 40 years old
Younger than 40 years old
Diabetes runs in family
Physically active
ra
ft
25
Reduced risk
D
Live in the North Island
_
Maori/Pacific Islander
1 Discuss which risk factors above are able to be influenced by an individual.
2 aWhich risk factor above is most associated with living in the North Island?
Explain why.
bWould it help someone who wanted to avoid diabetes to move to the South
Island?
3 The risk of an overweight person developing Type 2 diabetes is 10%.
Complete this sentence: ‘Compared to the general population, a person who is
overweight is about
times more likely to develop Type 2 diabetes.’
(Source: http://www.diabetes.org.nz/resources/DHB_figures)
9781442549487_CH-25.indd 384
6/13/11 4:03 AM
25 Further probability
385
9
Relative risk
Here we consider relative risk – that is, the likelihood of the event when given factors are taken into
account. The relative risk of an event for a particular group or to a certain kind of exposure describes
how likely this risk is compared to the risk of the same event for another group or to a different kind
of exposure.
Examples
1The relative risk of a smoker developing lung cancer compared to a non-smoker developing
lung cancer – here the two groups (smokers and non-smokers) are complementary.
2The relative risk of drowning in a swimming pool compared to drowning in any body of
water – here the first group is those who swim in a pool, and this is a subset of the second
group, which is those who swim anywhere, including a pool.
on
How do we calculate relative risk?
ar
s
Relative risk of an event for group A compared to the risk of the same event for group B is the ratio of
the probability of the event for group A to the probability of the event for group B.
ht
Pe
Example
This table shows data for a study of deaths from lung cancer versus other causes of death in the
United States. The table also shows whether the person was a smoker or not.
rig
(Source: http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/E/Epidemiology.html)
Non-smoker
397
Other causes
6919
4614
Total
7316
4651
ly
-
co
37
py
Smoker
Lung cancer
397 = 5.43%.
7316
Risk of a non-smoker dying of lung cancer: 37 = 0.80%.
4651
Based on this study, the relative risk of dying of lung cancer for a smoker compared to a ­
non-smoker dying of lung cancer, is 5.43 = 6.8.
0.80
25
D
ra
ft
on
Risk of a smoker dying of lung cancer:
That is, a smoker is almost seven times more likely to die of lung cancer than a non-smoker.
T
I
P
Relative risk is not a probability. Relative risk can have values greater than 1
(e.g. relative risk of cancer for a smoker compared to cancer for a non-smoker)
or less than 1 (e.g. risk of head injury for a cyclist who wears a crash helmet
compared to risk of head injury for all cyclists).
9781442549487_CH-25.indd 385
6/13/11 4:03 AM
386
2.12 Probability methods
EXERCISE
25.03
1 Two airlines flew the Auckland to
Christchurch route in December 2010 and
January 2011. This table shows the number
of on-time flights and the number of
delayed flights over that period.
Air
New Zealand Jetstar
Total
105
56
161
On-time
947
199
1146
1052
255
1307
Total
4 The table gives data regarding seat-belt use
and blood alcohol concentration (BAC) for
car and van drivers killed between 2005 and
2007. BAC is measured in mg of alcohol per
100 mL of blood.
on
Delayed
Calculate the relative risk of being delayed
on this route when flying Jetstar compared
to being delayed when flying Air New
Zealand.
Total
87
19
63
26
113
ly
44
50
on
Have not
had lessons
7
Pe
Calculate the relative risk of failing the
practical driving test if an applicant has
not had lessons from a professional driving
instructor compared to failing the test
if the applicant has had lessons from a
professional driving instructor.
D
43
ra
ft
25
Had lessons
Total
-
Pass test Fail test
3 The risk of a car being stopped for any
reason by the police when the driver is
using a mobile phone is 0.048. The risk of
a car being stopped for any reason when
the driver is not using a mobile phone is
0.000 32. What is the relative risk of a car
being stopped for any reason when the
driver is using a mobile phone compared to
the car being stopped when the driver is not
using a mobile phone?
9781442549487_CH-25.indd 386
Wearing a Not wearing
seat-belt
a seat belt
ht
co
py
2 A testing centre keeps statistics on whether
applicants for a driver licence pass the
practical test. The centre has also recorded
whether applicants have had lessons from a
professional driving instructor.
Car and van drivers killed 2005–07
Below the 80 mg
per 100 mL of blood limit
87%
13%
Above the 80 mg
per 100 mL
of blood limit
54%
46%
rig
ar
s
(Source: http://www.flightstats.com)
(Source: www.transport.govt.nz/research/.../Alcohol-DrugsCrash-Factsheet.pdf)
What is the relative risk for a driver not
wearing a seat-belt and who was over
the legal BAC limit of 80 mg per 100 mL
compared to the driver who was not
wearing a seat-belt and was below the
same limit?
5 This table shows data for flight delays and
numbers of flights operated from Auckland
to Brisbane over a two-month period.
Total
number
On-time Delayed of flights
Air New
Zealand
China Airlines
Emirates
Pacific Blue
Qantas
95
28
123
4
42
19
77
1
18
5
28
5
60
24
105
(Source: http://www.flightstats.com)
6/13/11 4:03 AM
25 Further probability
co
30+ years 20–29 years 15–19 years
March
8
April
9
May
11
June
14
July
13
August
13
September
12
October
12
ar
s
November
December
8
What is the relative risk of experiencing
a wet day in Wellington in the second
half of the year (July–December: 184 days
altogether) compared to the first half of the
year (January–June: 181 days altogether)?
Pe
10
8 The graph shows the relative risk of fatal
crash by blood alcohol level for different age
groups of driver.
1
3
30
2.9
8.7
50
5.8
17.5
30.3
200
80
16.5
50.2
86.6
160
age 20–29
140
age 30+
5.3
Relative risk of fatal crash by blood alcohol level
15
Current New Zealand BAC limit
ly
on
ra
ft
a How does the table show that the
calculation of risk is made in relation to
that of a sober driver aged 30+ years?
b Estimate the relative risk, at all age levels,
for a driver with a BAC of 50 compared
to a driver with a BAC of 30.
7 A ‘wet day’ is defined as one that has at
least 1.0 mm of rain. This table gives the
mean number of wet days in Wellington for
each month.
9781442549487_CH-25.indd 387
120
100
25
80
60
40
20
0
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
BAC (mg/100 mL)
( Source: www.transport.govt.nz/research/.../Alcohol-DrugsCrash-Factsheet.pdf)
age 15–19
180
D
7
 0
(Source: MOT Crash Fact Sheet 2008)
February
py
Age of driver
7
-
BAC
January
ht
6 This table shows calculations of relative
risk for being involved in a fatal road crash
by both age and blood alcohol concentration
(BAC). BAC is measured in mg of alcohol
per 100 mL of blood.
Wet days
rig
Increase in risk
Month
on
a Based on the data for this two-month
period, calculate these relative risks for
being delayed on a flight from Auckland
to Brisbane:
i flying Qantas compared to Pacific Blue
ii flying Emirates compared to Air New
Zealand
iii flying Pacific Blue compared to China
Airlines
iv flying Air New Zealand compared to
any other airline.
b Which airline has the lowest relative risk
of delay on this route compared to any
other?
c For which airline is the data most
unreliable? Explain.
387
a At what BAC is a teenage driver ten times
more likely to have a fatal road crash than
if the driver were completely sober?
b A driver aged 40 is almost twice the
legal BAC, which is 80 mg per 100 mL of
blood at the time of writing. How many
more times likely is this driver to have a
fatal road crash than if their BAC were
just on the legal limit?
6/13/11 4:03 AM
26 The normal distribution
419
2 Applications
ar
s
Pe
ht
First, enter the mean (m) and the standard deviation (s).
These values are used to generate a scale for the normal
distribution with those parameters. Note that the default
values (0, 1) give the standard normal curve.
on
The e-book version of this textbook comes
with the application ‘Pearson Normal
Probability Calculator’, which can
calculate any normal probability, given
the mean and standard deviation.
py
rig
Then choose one of these three options:
co
1 P(X < a)
on
ly
-
2 P(X > a)
ra
ft
3 P(b < X < c).
26
D
Enter the value of a, or values of b and c, then click on
‘Close’, and the application draws a normal curve and uses
colour to display the area that corresponds to the required
probability.
9781442549487_CH-26.indd 419
6/13/11 4:17 AM
421
26.06
-
EXERCISE
co
py
rig
ht
Pe
ar
s
on
26 The normal distribution
TECH
ra
ft
on
ly
1 X has a normal distribution with a mean of 36 and a standard deviation of 8.
Use appropriate technology to calculate these probabilities.
a P(X < 39)
b P(X < 27.5)
c P(X > 33)
d P(X > 41.74)
e P(39 < X < 51)
f P(33.6 < X < 39.2)
g P(28.9 < X < 43.5)
26
D
2 X has a normal distribution with a mean of 19 and a standard deviation of 1.73. Match each
probability below with the correct spreadsheet formula (A–C) from the given list.
a P(X < 21) b P(X > 21)
(A) =NORMDIST(19,21,1.73,TRUE)
(B) =NORMDIST(21,19,1.73,TRUE)
(C) =1-NORMDIST(21,19,1.73,TRUE)
3 X has a normal distribution with a mean of 19 and a standard deviation of 1.73. Match each
probability below with the correct spreadsheet formula (A–E) from the given list.
a P(18.9 < X < 23.4)
b P(X < 18.9 or X > 23.4)
(A) =NORMDIST(23.4,19,1.73,TRUE)+NORMDIST(18.9,19,1.73,TRUE)
(B) =NORMDIST(18.9,19,1.73,TRUE)+1-NORMDIST(23.4,19,1.73,TRUE)
(C) =NORMDIST(23.4,19,1.73,TRUE)+NORMDIST(21,19,1.73,TRUE)-1
9781442549487_CH-26.indd 421
6/13/11 4:18 AM
430
2.12 Probability methods
The area to the right of k is 0.325. Therefore the area to the left is 1 - 0.325 = 0.675.
Because the CAS
calculator works with
left-hand area, we enter
0.675 as the ‘area’.
That is,
P(X > 215.176) = 0.325.
DATA
ar
s
on
The steps for the Casio ClassPad 300 are explained in the CAS information sheet in
the Theta Mathematics Teaching Resource, and here is a hyperlink if you are using the
e-book version of this textbook.
ht
rig
py
ra
ft
First, enter the mean (m) and the standard deviation (s).
These values are used to generate a scale for the normal
distribution with those parameters.
D
26
on
ly
-
co
The e-book version of this textbook
comes with the application ‘Pearson
Normal Probability Calculator’, which
can calculate any value on a normal
distribution scale, given the mean,
standard deviation, and probability of
being either greater than or less than the
unknown value.
Pe
2 Applications
Then choose one of these two
options:
1 P(X < a) = k
2 P(X > a) = k
9781442549487_CH-26.indd 430
6/13/11 4:18 AM
432
2.12 Probability methods
-
co
py
rig
ht
Pe
ar
s
on
The screenshot shows the result of an
‘inverse normal’ calculation for a normal
distribution with a mean of 212 and a
standard deviation of 7.
ly
b
ra
ft
1 Each diagram shows a normal curve with
a mean of 10 and a standard deviation of
3, and an associated probability. Match
each diagram with the correct spreadsheet
formula (A–G) that would be used to
calculate the value(s) on the horizontal axis.
a
TECH
0.77
D
26
26.10
on
EXERCISE
10
k
c
0.85
j
=
10
l
(A) =NORMINV(0.85,10,3)
(B) =NORMINV(0.38,10,3)
9781442549487_CH-26.indd 432
(C) =NORMINV(0.77,10,3)
(D) =NORMINV(0.31,10,3)
(E) =NORMINV(0.15,10,3)
0.38
=
10
(F) =NORMINV(0.23,10,3)
(G) =NORMINV(0.89,10,3)
6/13/11 4:18 AM
27 Simulation methods
on
dNow recalculate the
spreadsheet at least
50 times. Write the number
of purchases needed to get
all five colours each time.
eExplain how you would
estimate the ‘average’
number of purchases
needed to qualify for a free
car-wash.
fInvestigate to find out
the number of purchases
needed if a sixth colour,
equally likely with the
others, were added.
ht
Pe
4 The World Series
The World Series is the name
given to the final stages of
the annual Major League Baseball Championships in North
America.
ar
s
443
co
py
rig
The winner of the World Series is the first team to record four
victories, which means the series is a ‘best of seven’ event. In
this simulation, assume that each team is equally likely to win
an individual game.
D
ra
ft
on
ly
-
The spreadsheet extract shows the result of simulating
20 different years of World Series games between team A and
team B. This simulation shows which team wins each game, and
also shows the cumulative number of wins at each stage. No more games are needed after scores
of 4–0, 4–1, 4–2, 4–3, 0–4, 1–4, 2–4 and 3–4. This is shown, in some cases, by blank cells in the
columns for games 5 and 6.
9781442549487_CH-27.indd 443
27
6/13/11 4:47 AM
460
2.14 Systems of equations
28
Simultaneous equations
Mathematics and Statistics in the New Zealand Curriculum
Mathematics: Patterns and relationships
Level 7
• M7-8 Form and use pairs of simultaneous equations
Pe
ht
A linear equation can be
represented on a graph by a
straight line.
co
py
rig
Simultaneous equations are equations that apply
at the same time. Here we consider a pair of
equations in two unknowns, x and y. The solution
will be the numbers x and y that satisfy both
equations. Note: if we have two unknowns, we need
two equations in order to solve for x and y.
We start with ‘linear’ equations in x and y. There
are several methods that can be used to solve these.
ar
s
on
Achievement Standard
Mathematics and Statistics 2.14 – Apply systems of equations in solving problems
Finding intersections by drawing lines
ra
ft
on
ly
-
Example
Draw a graph to solve these simultaneous equations.
(1) x + y = 5 (2) y = 2x − 1
Answer
y
5
D
y = 2x – 1
3
The graphs intersect at the point (2, 3).
x+y=5
Hence the solution is x = 2 and y = 3.
28
–1
EXERCISE
2
5 x
28.01
Find the point of intersection of these pairs of lines by drawing graphs.
1 y = 3x + 1
y = 2x + 2
3 2x + y = 4
y = x − 5
5 y + 3 = x
3x + y = 1
7 2x + y = 0
y = x − 3
2 y = 3 − x
y = x + 7
4 2x − y = 3
y = 5x
6 2x = y − 5
x + y = 2
8 y = x + 2
2y = 5x − 2
9781442549487_CH-28.indd 460
6/13/11 5:22 AM
28 Simultaneous equations
16 A business uses different depreciation
rates for different items of equipment. A
computer decreases in value by 30% per
year, and a photocopier decreases in value
by 20% per year.
The business buys one computer and one
photocopier for a total of $10 000. At the
end of the year, they are valued at $7600
altogether. How much did the business pay
for each machine?
17 An internet service provider offers users a
choice of two monthly rate plans.
on
ar
s
py
-
ly
40
D
y = 2x – 50
on
30
co
A
Hours on-line
ra
ft
Cost per month ($)
B
Pe
Suppose x is the number of hours on-line,
and y is the monthly charge. This graph (not
drawn to scale) is provided to users, and
shows the equations for Plan B.
y = 30
18 A school uses two minibuses to take a group
of students on a field trip.
• If one student moved from the first
minibus to the second, the two
minibuses would have the same number
of students.
• However, if a student moved from the
second minibus to the first, the first
minibus would have twice as many
students as the second minibus.
a Use this information to write a pair of
simultaneous equations.
b How many students are in the group
altogether?
ht
Each hour costs $1.50.
The first 40 (or fewer) hours
cost $30.
Hours over 40 cost $2 each.
a Show that both of the equations for
Plan B give the same result for a user
who expects to be connected for exactly
40 hours.
b Write the equation for Plan A.
c Solve some simultaneous equations to
find the range of values for which Plan B
is a better deal than Plan A. Show your
working, and explain what you are
working out at each step.
rig
Plan A
Plan B
The great picnic puzzle
PUZZLE
This puzzle was originally published in 1914
in Sam Loyd’s Cyclopedia of 5000 Puzzles, Tricks
and Conundrums with Answers.
When they started off on the great annual
picnic, every wagon in town was pressed into
service.
Half-way to the picnic ground, 10 wagons
broke down, so it was necessary for each
of the remaining wagons to carry one more
person.
When they started for home, it was discovered that 15 more wagons were out of
commission so, on the return trip, there were three persons more in each wagon than
when they started out in the morning.
Now, who can tell how many people attended the great annual picnic?
9781442549487_CH-28.indd 467
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Uncorrected Sample Pages
Theta Mathematics has been completely updated to reflect the
current requirements of Mathematics and Statistics in the New
Zealand Curriculum. In order to provide teachers and students with
a comprehensive package that covers all thirteen Achievement
Standards, Theta Mathematics is complemented by Theta Dimensions.
This edition follows in
the tradition of all David
Barton’s resources and
contains:
• material that addresses the requirements
of the front end of the New Zealand
Curriculum, including the Vision, Key
Competencies, Values and Cross and Bicultural references
• concise theory notes written with
students’ needs in mind
• comprehensive worked examples that are
well set out
• plenty of questions in context
• graded exercises to practise skills and
build solid foundations
• questions that require understanding and
explanation, and some extended working
• investigations and puzzles, many new, to
motivate students and stimulate thinking
• references to technology throughout,
including animations, spreadsheet work,
CAS calculators, and websites that are
linked from www.mathematics.co.nz
• complete answers.
Also available
THETA MATHEMATICS
WORKBOOK
• Includes the topics that lead
on to either a Statistics course
or a Calculus course in Year 13
• Includes the topics that are
currently assessed externally.
THETA DIMENSIONS and
THETA DIMENSIONS WORKBOOK
• Includes the internally assessed topics that lead into a
Statistics course in Year 13.
ISBN 978-1-4425-4948-7
9
9781442549487_3 climbers_COV.indd 1
781442 549487
MATHEMATICS
The chapters for each Achievement Standard are colour-coded to
make them easy to find, and liberal use of colour throughout the book
makes it easy and interesting to use. Helpful tags on the edge of each
page enable students to navigate between the answer section and the
related exercises.
Theta Mathematics
2.1
Co-ordinate geometry
2.2
Graphical models
2.4
Trigonometric relationships
2.6
Algebraic methods
2.7
Calculus methods
2.12 Probability methods
2.13 Simulation
2.14 Systems of equations