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978-1-107-67670-1 - CambridgeMaths: NSW Syllabus for the Australian Curriculum: Year 10: Stage 5.1/5.2/5.3
Stuart Palmer, David Greenwood, Sara Woolley, Jenny Goodman and Jennifer Vaughan
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Cambridge University Press
978-1-107-67670-1 - CambridgeMaths: NSW Syllabus for the Australian Curriculum: Year 10: Stage 5.1/5.2/5.3
Stuart Palmer, David Greenwood, Sara Woolley, Jenny Goodman and Jennifer Vaughan
Excerpt
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Chapter
1
Measurement
What you will learn
1A
1B
1C
1D
1E
1F
1G
1H
1I
Converting units of measurement
Accuracy of measuring instruments
Pythagoras’ theorem in three-dimensional problems
Area of triangles, quadrilaterals, circles and sectors Surface area of prisms and cylinders
Surface area of pyramids and cones
Volume of prisms and cylinders
Volume of pyramids and cones
Volume and surface area of spheres
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Cambridge University Press
978-1-107-67670-1 - CambridgeMaths: NSW Syllabus for the Australian Curriculum: Year 10: Stage 5.1/5.2/5.3
Stuart Palmer, David Greenwood, Sara Woolley, Jenny Goodman and Jennifer Vaughan
Excerpt
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3 nSW Syllabus
for the australian
Curriculum
Strand: Measurement and Geometry
Substrands: nuMBERS oF anY MaGnItuDE
aREa anD SuRFaCE aREa
VoluME
Outcomes
A student interprets very small and very large units of measurement, uses scientifi c notation, and rounds to signifi cant fi gures.
(MA5.1–9MG)
A student calculates the areas of composite shapes, and the surface areas of rectangular and triangular prisms.
Monolithic domes
Monolithic domes are round one-piece structures with a smooth spherical-like surface. They offer excellent protection from earthquakes, bushfi res and cyclones because of their shape and durability. They are extremely energy effi cient because of the minimal surface area for the volume contained within the structure. The fi rst type of monolithic domes used were igloos, which are very strong and provide good insulation in freezing conditions. The minimal surface area of the dome means that there is less surface for heat to be transferred to the outside air. A cube, for example, containing the same volume of air as the dome has about 30% more surface area exposed to the outside air. Volumes and surface areas of spheres and other solids can be calculated using special formulas.
(MA5.1–8MG)
A student calculates the surface areas of right prisms, cylinders and related composite solids.
(MA5.2–11MG)
A student applies formulas to fi nd the surface areas of right pyramids, right cones, spheres and related composite solids.
(MA5.3–13MG)
A student applies formulas to calculate the volumes of composite solids composed of right prisms and cylinders.
(MA5.2–12MG)
A student applies formulas to fi nd the volumes of right pyramids, right cones, spheres and related composite solids.
(MA5.3–14MG)
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Cambridge University Press
978-1-107-67670-1 - CambridgeMaths: NSW Syllabus for the Australian Curriculum: Year 10: Stage 5.1/5.2/5.3
Stuart Palmer, David Greenwood, Sara Woolley, Jenny Goodman and Jennifer Vaughan
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Chapter 11Measurement
Chapter
Measurement
pre-test
4
1 Evaluate the following.
a 2 × 100
d 0.043 × 1002
b 5 ÷ 102
e 62 900 ÷ 1000
2 Evaluate the following.
a A = × b when = 3 and b = 7
b A = b × h when b = 10 and h = 3
1
c A = b × h when b = 2 and h = 3.8
2
1
d A = h(a + b) if a = 2, b = 3 and h = 4
2
3 Find the perimeter of these shapes.
a
b
c 230 ÷ 102
f 1.38 × 10002
1m
3 cm
c
3 cm
5 cm
3m
4 Use these rules to find the circumference (C)) and area ((A) of the circles, correct to
1 decimal place: C = 2prr and A = prr2, where r is the radius.
a
b
c
2m
3.9 km
12 cm
5 Use Pythagoras’ theorem a2 + b2 = c2 to find the value of the pronumeral in these triangles.
Round to 1 decimal place where necessary.
a
b
c
12
2
c
3
1
c
a
13
4
6 Find the surface area and volume of these solids. Round to 1 decimal place
where necessary.
a
b
c
6 cm
4 cm
2m
2 cm
5 cm
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978-1-107-67670-1 - CambridgeMaths: NSW Syllabus for the Australian Curriculum: Year 10: Stage 5.1/5.2/5.3
Stuart Palmer, David Greenwood, Sara Woolley, Jenny Goodman and Jennifer Vaughan
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5 Measurement and Geometry
1A Converting units of measurement
Stage
5.3#
5.3
5.3§
5.2
5.2◊
5.1
4
The international system of units (SI) is from the
French Système International d’Unités that has been
adapted by most countries since its development in the
1960s. It is called the Metric system.
As our technology expands and our knowledge of
the universe widens, the need increases for even larger
and smaller units of measurements to exist. The common
prefixes of milli ( 1 th), centi ( 1 th) and kilo (1000)
1000
100
are joined by others. Giga (109) and nano (10-9) are just
two of the prefixes added to our everyday usage.
let’s start: Powers of 10
■■
Metric prefixes in everyday use
prefix
tera
giga
mega
Symbol
T
G
M
Factor of 10
1 000 000 000 000
1 000 000 000
1 000 000
Standard form
1012
1 trillion
9
1 billion
6
1 million
3
10
10
kilo
k
1 000
10
1 thousand
hecto
h
100
102
1 hundred
deca
da
10
10
1 ten
deci
d
10-1
1 tenth
0.1
-2
centi
c
0.01
10
1 hundredth
milli
m
0.001 10-3
1 thousandth
micro
nano
µ
n
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0.000001 0.000000001
-6
1 millionth
-9
1 billionth
10
10
Key ideas
In your workbook, write each of the following as a
power of 10. The first line has been done for you.
10 = 101
0.1 = 10-1
100
0.01
1000
0.001
10 000
0.0001
100 000
0.00001
100 000
0.000001
1 000 000
0.0000001
100 000 000
0.00000001
1 000 000 000
0.000000001
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Cambridge University Press
978-1-107-67670-1 - CambridgeMaths: NSW Syllabus for the Australian Curriculum: Year 10: Stage 5.1/5.2/5.3
Stuart Palmer, David Greenwood, Sara Woolley, Jenny Goodman and Jennifer Vaughan
Excerpt
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6
Chapter 1 Measurement
Example 1 Converting units of time
Convert 3 minutes to:
a microseconds (µs)
b
nanoseconds (ns)
SolutIon
ExplanatIon
a 3 minutes = 180 s
= 180 × 106 s
= 1.8 × 108 s
1 min = 60 s
1 s = 1 000 000 µs
Express the answer in scientific notation.
b 3 minutes = 180 s
= 180 × 109 ns
= 1.8 × 1011 ns
1 min = 60 s
1 s = 1 000 000 000 (109) ns
Example 2 Converting units of mass
Convert 4 000 000 000 milligrams (mg) to tonnes (t).
SolutIon
ExplanatIon
4 000 000 000 mg = 4 000 000 g
= 4000 kg
=4t
mg means milligrams
1000 mg = 1 g
1000 g = 1 kg
1000 kg = 1 t
Dividing by the conversion factor converts a small unit to a
larger unit as there are fewer of them.
MA
d Mt
h dm
R
HE
T
1 What is meant by each of the following abbreviations?
a ms
b mm
c km
e Mg
f mg
g ns
U
R K I NG
C
F
PS
Y
WO
LL
Exercise 1A
M AT I C A
2 Write down what is meant by each of the following expressions.
a an income of 40K
b an 8 GB USB drive
c 2 Mt of sand
d 3 ns
e 40 kB of digital data
3 Complete this flow diagram, which could be used to convert units of time.
× 24
day
hour
min
s
ms
ms
ns
÷ 1000
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Cambridge University Press
978-1-107-67670-1 - CambridgeMaths: NSW Syllabus for the Australian Curriculum: Year 10: Stage 5.1/5.2/5.3
Stuart Palmer, David Greenwood, Sara Woolley, Jenny Goodman and Jennifer Vaughan
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7 Measurement and Geometry
R
T
HE
TB
GB
MB
kB
R K I NG
C
F
PS
LL
U
Y
WO
MA
4 Complete this flow diagram, which could be used to convert between measurements of digital
information. (Note: B stands for ‘byte’.)
M AT I C A
B
R
M AT I C A
WO
U
MA
6 The Hatton family have a monthly internet and pay TV 5 GB home bundle, costing
$219.12 per month.
a What is the cost over the 24-month contract?
b What is the monthly cost for the bundle per gigabyte?
c What is the average daily allowance for this bundle?
R
T
HE
F
PS
R K I NG
C
F
PS
Y
T
HE
C
LL
MA
5 Complete these conversions.
Examples 1 & 2 a 2000 t = _______________________ Mt
b 5 s = ________________ ms
c 5 km = _______________ m
d 20 dm = _____________ m
e 10 hectometres = ________________ km
f 3 MB = ________________ B
g 32 GB = _______________ MB = ______________ B
h 60 mm = _____________ m
i 200 cm = ______________ m
j 5 kg = ______________ g
k 0.06 g = ____________ mg
l 1 t = _____________ g
2
m 1 s = ______________ milliseconds (ms)
n 1 s = _____________ microseconds (µs)
o 2 terabytes (TB) = ___________________ GB
p 35 mg = _______________ g
q 2 × 1030 kg = ____________ Mt
r 1 GHz (gigahertz ) = _______________ Hz
s 1 TB = _______________ GB = ____________ MB = ____________________ kB
t 4 hectometres = _________ m
R K I NG
LL
U
Y
WO
M AT I C A
7 Olympic swimmer Michael Phelps won a gold medal at the 2012 London Olympic Games for the
200 m medley in a time of 1:54.27. What does this mean and what is this time when converted to
milliseconds?
8 In the Olympic Games, the time-keeping devices are said to measure correct to the nearest
thousandth of a second. Which SI prefix is used for this purpose?
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978-1-107-67670-1 - CambridgeMaths: NSW Syllabus for the Australian Curriculum: Year 10: Stage 5.1/5.2/5.3
Stuart Palmer, David Greenwood, Sara Woolley, Jenny Goodman and Jennifer Vaughan
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1A
WO
9 How many milliseconds are there in one day? Express this value in scientific notation.
C
R
MA
HE
T
10 The chemical element copernicium 277 (277Cn) has a half-life of 240 microseconds.
How many seconds is this?
R K I NG
U
F
PS
Y
Chapter 1 Measurement
LL
8
M AT I C A
11 The average human eye blink takes 350 000 microseconds. How many times can the average person
blink consecutively, at this rate, in 1 minute?
MA
12 How many times greater is the prefix M to the prefix m?
R
T
HE
13 Complete this comparison:
1 microsecond is to 1 second as what 1 second is to _______ days.
R K I NG
C
F
PS
Y
U
LL
WO
M AT I C A
14 Stored on her computer, Hannah has photos of her recent weekend away. She has filed them according
to various events. The files have the following sizes: 1.2 MB, 171 KB, 111 KB, 120 KB, 5.1 MB and
2.3 MB. (Note that some computers use KB instead of kB in their information on each file.)
a What is the total size of the photos of her weekend, in kilobytes?
b What is the total in megabytes?
c Hannah wishes to email these photos to her mum. However, her mum’s file server can only receive
email attachments no bigger than 8 MB. Is it possible for Hannah to send all of her photos from
the weekend in one email?
15 Complete the tables for these time conversion scales.
a 1 second is equivalent to:
b A year containing 365 days is equivalent to:
millisecond millisecond microsecond
microsecond
nanosescond
nanosescond
minute
second
hour
minute
day
hour
week
day
month
week
year
month
century
century
millennium
millennium
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978-1-107-67670-1 - CambridgeMaths: NSW Syllabus for the Australian Curriculum: Year 10: Stage 5.1/5.2/5.3
Stuart Palmer, David Greenwood, Sara Woolley, Jenny Goodman and Jennifer Vaughan
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Measurement and Geometry
Enrichment: Half-life
16 The half-life of an isotope (a form) of a chemical
element is the time taken for half of its atoms to decay
into another form. Different isotopes have different
half-lives. They can range from billions of years to
microseconds. The table shows some isotopes and
their half-lives.
a Place the isotopes listed in the table in order from
the shortest half-life to the longest.
b How many times greater is the half-life of 24Na to
that of:
i 216Po?
ii 15O?
c If, initially, there is 20 mg of gold-198 (198Au),
how many days does it take for less than 1 mg to
remain?
d Research half-lives for other isotopes of elements
in the periodic table.
Isotope
Half-life
41Ar)
argon-41 (
1.827 hours
barium-142 (142Ba)
10.6 minutes
calcium-41 (41Ca)
130 000 years
carbon-14 (
14C14)
51Cr)
chromium-51 (
gold-198 (
198Au)
5730 years
27.704 days
2.696 days
hydrogen-3 (3H)
12.35 years
oxygen-15 (15O)
122.24 seconds
213Po)
4.2 microseconds
216Po)
0.15 seconds
polonium-213 (
polonium-216 (
sodium-24 (24Na)
15 hours
232
72 years
uranium-232 (
69
zinc-69 ( Zn)
U)
57 minutes
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978-1-107-67670-1 - CambridgeMaths: NSW Syllabus for the Australian Curriculum: Year 10: Stage 5.1/5.2/5.3
Stuart Palmer, David Greenwood, Sara Woolley, Jenny Goodman and Jennifer Vaughan
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10
Chapter 1 Measurement
1B accuracy of measuring instruments
Humans and machines measure many different
things, such as the time taken to swim a race, the
length of timber needed for a building and the
volume of cement needed to lay a concrete path
around a swimming pool. The degree or level of
accuracy required usually depends on the intended
purpose of the measurement.
All measurements are approximate. Errors can
happen as a result of the equipment being used or
the person using the measuring device.
Accuracy is the measure of how true the
measure is to the ‘real’ one, whereas precision
is the ability to obtain the same result over and
over again.
Stage
5.3#
5.3
5.3§
5.2
5.2◊
5.1
4
During many sporting events, a high degree of accuracy is required for time-keeping.
let’s start: Rounding a decimal
1 A piece of timber is measured to be 86 cm, correct to the nearest centimetre.
a What is the smallest decimal that it could be rounded from?
b What is the largest decimal that is recorded as 86 cm when rounded to the nearest whole?
2 If a measurement is recorded as 6.0 cm, correct to the nearest millimetre, then:
a What units were used when measuring?
b What is the smallest decimal that could be rounded to this value?
c What is the largest decimal that would have resulted in 6.0 cm?
Key ideas
3 Consider a square with sides of length 7.8941 cm.
a What is the perimeter of the square if the side length is:
i left with 4 decimal places?
ii rounded to 1 decimal place?
iii truncated (i.e. chopped off) at 1 decimal place?
b What is the difference between the perimeters if the decimal is rounded to 2 decimal places or
truncated at 2 decimal places or written with two significant figures?
■■
The limits of accuracy tell you what the upper and lower boundaries are for the true
measurement.
– Usually, it is ±0.5 × the smallest unit of measurement.
For example, when measuring to the nearest centimetre, 86 cm has limits from 85.5 cm up to
(but not including) 86.5 cm.
Anything in this zone
would be 86 cm, to the
nearest centimetre.
86.5 is rounded to 87.
85.5
85
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86
87
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