9
MEASUREMENt
geometry of
the earth
Did you know that the sun rises in Australia before most of the rest of the world, and that in the
USA it is still ‘yesterday’? The only places that are ‘ahead’ of us in time are the Pacific islands such
as New Zealand, Fiji, Vanuatu and Tonga. To understand why, we must study the geometry and
geography of the world, taking into account that the Earth is a sphere rather than a flat surface.
Chapter outline
MM6 9-01
Latitude and longitude
MM6 9-02
Great circle distances
MM6 9-03
Longitude and time differences
MM6 9-04
International time zones
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In this chapter you will:
Shutterstock.com/Merydolla
• use latitude and longitude to locate positions on the Earth’s surface
• use the arc length formula to calculate the great circle distance between two places on the
Earth’s surface
• use the difference in longitude between two places on Earth to calculate the difference between
their local times
• use international and Australian time zones to calculate the local time of a place
• solve travel problems involving time zones, including daylight saving and the International
Date Line
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Crossword of
the Earth
tERMiNology
angulardistance
coordinates
great circle
International Date Line
longitude
prime meridian
arclength
daylightsaving
Greenwich Meridian
kilometre
meridian
small circle
CoordinatedUniversalTime(UTC)
Equator
hemisphere
latitude
parallel
time zone
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Assignment 9
SkillCheck
1
a
Write the coordinates of points A and B.
b
What is the length of the interval AB ?
c
Write the coordinates of point C if it is
3 units up and 4 units left from B.
y
5
4
3
2
A
–3 –2 –1 0
–1
2
3
a
Plot the points D(3, 5) and E(3, 2) on a number plane.
b
What is the length of the interval DE?
B
1
1
2
3
4 x
X
Find, correct to two decimal places, the
length of the arc XY in the diagram.
8 cm
O 96°
8 cm
4
5
6
a
Calculate his average speed.
b
How long will it take him to travel 50 km at this speed? Answer in hours
and minutes.
What is the time:
a
7 hours after 7:00 p.m.?
b 8 hours before 2:00 p.m.?
c
11 h 18 min after 1:15 p.m.?
d 7 h 36 min before 2:20 a.m.?
Use the
a
7
key on your calculator to convert these times to hours and minutes.
22.7 hours
b 6.3 hours
Convert these times to hours and minutes.
a
384
Y
A marathon runner travels 42 kilometres in 3 12 hours.
482 minutes
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b 271 minutes
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9-01 latitude and longitude
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When describing the location of a point on a number plane or map, we use a coordinate
systeminvolvingorderedpairs(x, y).
–2
Rd
King St
Church
Madeline
a St
Sta n le
d
nR
Australian
coordinates
Foss St
H u nt leys P
o
a Rd
ri
Victo
Parramatta
River
HUNTLEYS
POINT
A
P has coordinates (3, 1) on this number plane.
int
1
–5
Fe
rn
yS
t
2
–3
–4
wS
St
Alexandr
St
Mount St
Geo
e
Cullens Ln
Herberto
n Ave
t
rge S
3
North Pd
Daram St
ne Rd
Mount St
Hunters Hill
Girls High
School
Burns Bay
4
Rd
Rd
3
Rd
2
ue
en
Av
1
5 x
Latitude and
longitude
Re
iby
Gladesto
P (3,1)
1
Jouber t St
d
2
Burns Bay Rd
d
nR
il S t
ga
fi
ne
R
de
Ry
Rd
Figtree Park
4
wS
Martin St
Figtree
Short St
3
–5 –4 –3 –2 –1
–1
Ab
t
i
es S
Jam
5
4
Bo n
6
y
5
Map of the
world
B
C
D
E
F
Hunters Hill Girls High School has coordinates D4 here.
PositionsontheEarth’ssurfacearedescribedbyacoordinatesysteminvolvinglatitude and
longitude.However,becausetheEarthisasphere,weuseaspecialgridoflinesthatrun
across and down a sphere. The diagrams below show this grid on a world globe and a flat
world map.
°E
60
E
30°
W
60°N
40°N
0°
20°N
0°
20°S
20°S
40°S
60°S
Degrees latitude
60°N
40°N
20°N
30
°
0°
40°S
180°W
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120°W
0°
60°W
60°E
Degrees longitude
9. Geometry of the Earth
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180°E
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Great and small circles
If you cut a ‘slice’ through a sphere, its shape is a circle. A slice through the centre of a
sphere is called a great circle, and its radius is the same as that of the sphere. Any other
slice is called a small circle, because its radius is smaller than that of a great circle.
Great circles
Small circles
Parallels of latitude
Parallels of latitude are imaginary parallel circles that run around the Earth. The largest
parallel of latitude is the Equator, 0°, which is a great circle. The other parallels of latitude
are all small circles. The North Pole has a latitude 90° north (90°N). The South Pole has a
latitude of 90°S.
North pole
90°N
75°N
60°N
45°N
30°N
15°N
0°
Equator
15°S
30°S
45°S
60°S
South pole
90°S
Northern
hemisphere
Southern
hemisphere
The angle of latitude is the angle the parallel makes with the Equator at the centre, O, of
the Earth. The diagrams below show the 50°N and 35°S parallels of latitude.
90°N
50°N
O
Angle of
latitude
Angle of
latitude
O
50°
35°S
Equator
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35°
Equator
90°S
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Meridians of longitude
15°E
30°E
45°W
Western
hemisphere
45°E
60°W
90°W
Prime
meridian 0°
meridian 0°
Prime
The angle of longitude is the angle the meridian
makes with the prime meridian at the centre, O, of the Earth.
The diagram shows the 35°E meridian of longitude.
Axis
S
35°E m idian
er
Greenwich is pronounced ‘gren-itch’
N
15°W
30°W
Eastern
hemisphere
Meridians of longitude are imaginary semicircles
that run down the Earth. They are ‘half’ great circles that
meet at the North and South Poles. The main meridian of
longitude is the prime meridian, 0°. It is also called
the greenwich meridian since it runs through the
RoyalObservatoryatGreenwichinLondon,England.
O
35°
0°
Angle of
longitude
Meridians of longitude range from 180°E to 180°W. 180°E
and 180°W are actually the same meridian, on the opposite
side of the Earth to the prime meridian. It runs through the
Pacific Ocean, east of Fiji.
N
EASTERN
HEMISPHERE
Hawaii
INTERNATIONAL
DATE LINE
10º N
EQUATOR
Kiribati
Samoa
Fiji
Tahiti
Tonga
Australia
30º S
New
Zealand
160ºE
0º
10º S
Pacific Ocean
180º
position coordinates
160º W
WESTERN
HEMISPHERE
140º W
LocationsontheEartharedescribedusinglatitude(°Nor°S)andlongitude(°Eor°W).For
example,Canberrahascoordinates(35°S,149°E),meaningitis35°southoftheEquator
and 149° east of the prime meridian.
0° prime
meridian
149°
0° Equator
0° Equator
35°S
Canberra
Canberra
0° prime meridian
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These diagrams show some mnemonics for understanding latitude and longitude.
‘a’ for
across
lat is fat or flat
Latitude
N°, S°
rings.
small circles
‘o’ for
orange, onion
Longitude
lines are
long
E°, W°
hoops.
great circles
Example 1
Match the following coordinates to the points illustrated.
a (50°S, 55°E)
b (30°N, 55°E)
c (75°N, 0°E)
d (0°, 0°)
e (0°, 55°E)
f (75°N, 55°E)
75°N
prime
meridian 0°
30°N
S
R
O
55°
T
Solution
Q
P
V
0°
U
W
50°S
Since ∠TOU = 55°, QSUW is the 55°E meridian of longitude.
a (50°S, 55°E) is W
b (30°N, 55°E) is S
c (75°N, 0°E) is P
d (0°, 0°) is T
e (0°, 55°E) is U
f (75°N, 55°E) is Q
Example 2
Sydney’s coordinates are (34°S, 151°E) while Tokyo’s are (35°N, 139°E).
a Find their difference in latitude.
388
b
Find their difference in longitude.
c
Which city is further west?
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Solution
Draw a rough sketch to position the cities.
139°E
0°
157°E
35°N
Tokyo
0°
34°S
Sydney
a Difference in latitude = 35° + 34° = 69°
Difference between 35°N and 34°S.
b Difference in longitude = 151° − 139° = 12°
Difference between 151°E and 139°E.
c
Sydney is further east, so Tokyo is further west.
Example 3
Goulburn has coordinates (34°S, 149°E). If Armidale is 4° north and 2° east of
Goulburn, what are its coordinates?
Solution
149°E
0°
151°E
Latitude = 34°S − 4° = 30°S
Longitude = 149°E + 2° = 151°E
The coordinates of Armidale are (30°S, 151°E).
0°
30°S
Armidale
34°S
Goulburn
Exercise 9-01 Latitude and longitude
Note: An atlas and/or a world globe may be helpful for this exercise.
1 Match each set of coordinates to a point on the Earth.
a (0°, 100°E)
b (40°S, 60°E)
c (60°N, 60°E)
d (0°, 60°E)
e (15°S, 100°E)
f (60°N, 100°E)
g (15°S, 60°E)
h (40°S, 100°E)
60°E
A
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1
100°E
meridian
D
F
G
9. Geometry of the Earth
60°N
B
O
40°
C
E
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Example
Equator
15°S
40°S
H
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2
Match each city to a letter on the map below.
a
Moscow (55°N, 37°E)
b Edinburgh (56°N, 3°W)
c
Athens (38°N, 23°E)
d San Francisco (38°N, 122°W)
e
Perth (32°S, 116°E)
f Beijing (40°N, 116°E)
g
Alexandria (31°N, 30°E)
h Tokyo (35°N, 139°E)
i
St Petersburg (60°N, 30°E)
j Johannesburg (26°S, 28°E)
k
Lima (12°S, 77°W)
l Hong Kong (22°N, 114°E)
C
D
E
H
K
180°W
3
J
60°W
0°
60°N
F
L
40°N
I
20°N
0°
20°S
A
60°E
120°E
40°S
180°E
Write the coordinates of each of the points on
the Earth.
a
V
b
T
c
U
d
Z
e
R
f
W
g
X
h
S
110°W
75°W
Are these small circles or great circles?
6
7
Y
T
45°
W
30°
Equator
Z
40°W
5
b
V
X
What can be found at the 90°S parallel of latitude?
the 30°N parallel
R S
U
4
a
the prime meridian
c
the 145°W meridian
Calculate the difference in latitude between each pair of cities, and state whether the
second city listed is north or south of the first.
a
Shanghai, China (31°N, 121°E) and New York, USA (40°N, 64°W)
b
Nairobi, Kenya (1°S, 37°E) and Bangkok, Thailand (13°N, 100°E)
c
Moscow, Russia (55°N, 37°E) and London, UK (51°N, 0°)
d
Auckland, New Zealand (37°S, 174°E) and Canberra (35°S, 149°E)
e
Melbourne (37°S, 145°E) and Cairo, Egypt (30°N, 31°E)
f
Newcastle (33°S, 151°E) and West Wyalong (34°S, 147°E)
a
Deniliquin is 1° due south of Hay. If Hay’s coordinates are (34°S, 145°E), what are
Deniliquin’s coordinates? Select A, B, C or D.
A
390
120°W
B
G
(34°S, 144°E)
B
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(35°S, 145°E)
C
(33°S, 145°E)
D
(34°S, 146°E)
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b Lord Howe Island is 6° due east of Forster. If Forster’s coordinates are (32°S, 152°E)
what are Lord Howe Island’s coordinates? Select A, B, C or D.
A (32°S, 146°E) B (38°S, 152°E) C (32°S, 158°E) D (26°S, 158°E)
8 Calculate the difference in longitude between each pair of cities, and state whether the
second city listed is east or west of the first.
Example
2
a Budapest, Hungary (47°N, 19°E) and Miami, USA (25°N, 80°W)
b Athens, Greece (38°N, 23°E) and Paris, France (49°N, 2°E)
c Havana, Cuba (23°N, 82°W) and Mexico City, Mexico (19°N, 99°W)
d Buenos Aires, Argentina (34°S, 58°W) and Johannesburg, South Africa (26°S, 28°E)
e Manila, Philippines (14°N, 121°E) and Port Moresby, Papua New Guinea (9°S, 147°E)
f
Finley (35°S, 145°E) and Bourke (30°S, 146°E)
9 Broken Hill is 7° due west of Dubbo. If Dubbo’s position is (32°S, 148°E), what is
Broken Hill’s position?
Example
3
10Ipswich is 2° north and 2° east of Moree (29°S, 150°E). What are Ipswich’s coordinates?
11Ballarat is 2° south and 6° west of Batemans Bay (35°S, 150°E). What are the coordinates
of Ballarat? Select A, B, C or D.
A (33°S, 144°E)
B (37°S, 156°E)
C (33°S, 156°E)
D (37°S, 144°E)
12aIn Australia, do Sydney and Tamworth lie on the same line of latitude or longitude?
b Do Athens, Greece and Sicily, Italy, lie on the same line of latitude or longitude?
13Where on Earth is it possible to stand on every meridian of longitude?
investigation
Atlas activities
Use an atlas or the Internet to locate the positions of different places around the world.
Some websites give the coordinates of any city you type in, for example, Google Earth
lists the coordinates at the bottom of the page.
1
Use a world map or globe to estimate the coordinates of each city.
a Washington DC, USA
bSingapore
c Adelaide, Australia
d Dublin, Ireland
e Lisbon, Portugal
f Moscow, Russia
2
Find the actual coordinates of the cities from question 1, and then list them in order from:
a west to east
3
b north to south
Which city is located at:
a (19°N, 155°W)?
b (14°N, 121°E)?
c (34°S, 58°W)?
d (42°N, 12°E)?
e (12°S, 131°E)?
f (42°N, 87°W)?
4
How does a flat map distort the positions and sizes of places with high latitudes?
5
Investigate where these regions are in the world and how they got their names:
a the Middle East
b the Orient
c Western world
d the Antipodes
eEcuador
f the West Indies
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9-02 great circle distances
In Chapter 3, we learnt the formula for the arc length of a circle.
SUMMARy
Arc length of a circle
l=
θ
× 2π r
360
r
l
θ
r
where θ is the number of degrees in the central angle, r is the radius and l is the length of
the arc on the circumference.
On a flat surface, the shortest distance between two points is
astraightline.SincetheEarth’ssurfaceiscurved,theshortest
distance between A and B is the arc length AB of the great circle
that passes through A and B. This is called the great circle
distance, and the size of angle ∠AOB, where O is the centre
of the Earth, is called the angular distance.
A
B
O
ExAMplE 4
A
O
63°
00
64
In the diagram on the right, A and B on the Earth’s surface
have an angular distance of 63°. Calculate the great circle
distance between A and B, correct to the nearest kilometre.
The radius of the Earth is 6400 km.
km
Solution
B
θ
× 2π r
360
63
=
× 2 × π × 6400
360
AB
B=
= 7037.1675…
≈ 7037 km
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ExAMplE 5
Beijing, China and Perth, Australia have coordinates (40°N, 116°E) and (32°S, 116°E)
respectively.
a
What great circle joins Beijing and Perth?
b
What is the angular distance between these two cities?
c
Hence, calculate the shortest distance between Beijing and Perth, to the nearest
kilometre, given that the Earth’s radius is 6400 km.
Solution
a
Draw either a rough flat map diagram or global diagram showing the positions of
Beijing and Perth.
The great circle joining Beijing and Perth is the 116°E meridian of longitude.
Beijing
0°
Beijing
40°N
40°N
0°
0°
Perth
0°
b
c
32°S
32°S
Perth
116°E
Angular distance = 40° + 32° = 72°
θ
Distance =
× 2π r
360
72
=
× 2π × 6400
360
= 8042.4772…
≈ 8042 km
SUMMARy
The radius of the Earth is approximately 6400 km.
Exercise 9-02 great circle distances
1
In the diagram below, O is the centre of the Earth and the
angular distance between P and Q on the Earth’s surface
is 166°. Calculate, correct to the nearest kilometre, the arc
length PQ.
Example
Q
O
4
166°
P
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Example
5
2a What great circle passes through C and D?
50°W
15°E
b What is the angular distance between C and D?
48°N
c Calculate the shortest distance between C and D to the
nearest kilometre.
A
d What great circle passes through A and D?
D
O
B
0°
C
e Calculate the great circle distance between A and D
to the nearest kilometre.
f
What is the distance between B and C, to the nearest kilometre?
3 Paris, France has coordinates (49°N, 2°E). What is its distance from the Equator in
kilometres? Select A, B, C or D.
A223
B980
C5250
D5473
4 Melbourne (37°S, 145°E) and Cairns (17°S, 145°E) lie on the same meridian of longitude.
a What is the angle made by these two cities at the centre of the Earth?
b Calculate the shortest distance between Melbourne and Cairns to the nearest
kilometre.
5 The Antarctic Circle near the South Pole is the 66.5°S parallel of latitude. Calculate
how many kilometres this circle is from the South Pole. Select A, B, C or D.
A123
B267
C2625
D7428
6 The Tropic of Capricorn is the 23.5°S parallel of latitude. Calculate, correct to the
nearest kilometre, how far this circle is from:
a the North Pole
b the Antarctic Circle.
7 Calculate, to the nearest kilometre, the shortest distance between each pair of cities.
a Osaka, Japan (34°N, 135°E) and Alice Springs (23°S, 135°E)
b Pretoria, South Africa (25°S, 28°E) and Minsk, Belorussia (54°N, 28°E)
c Kampala, Uganda (0°, 32°E) and Macapa, Brazil (0°, 51°W)
d Mawson, Antarctica (67°S, 63°E) and Chelyabinsk, Russia (54°N, 63°E)
e Port Moresby, Papua New Guinea (9°S, 147°E) and Nyngan (31°S, 147°E)
f
Leeton (34°S, 146°E) and Townsville (19°S, 146°E).
8 If you dug a hole from Beijing, China (40°N, 116°E) through the centre of the Earth and
came out on the other side, you would end up at point X, somewhere in Argentina.
a What would be the coordinates of the point X?
b How many kilometres are there between Beijing and point X:
i
in a straight line through the Earth’s centre?
ii along a great circle on the Earth’s surface?
9 The great circle distance between Sydney and Honolulu, Hawaii is 4170 km. What angle
is made at the centre of the Earth by these two cities? Answer to the nearest degree.
10It is possible to fly from Perth (32°S, 115°E) to Tierra del Fuego (54°S, 65°W) on
the southern tip of South America by going directly over the South Pole along a line
of longitude. Calculate the great circle distance between Perth and Tierra del Fuego,
correct to the nearest kilometre.
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tEChNology: loCAtioNS oN thE EARth
Distance calculator websites
Use a distance calculator website to enter the names or coordinates of two places in the
world, then find the shortest distance between them. Some computers have a distance
calculator included in their system software.
global positioning system
A global positioning system (GPS) or satellite navigator uses Earth satellites to calculate
a person’s location on Earth. Investigate how GPS works. Find out what an EPIRB is and
how it is used in navigation.
9-03 longitude and time differences
When the Sun shines
directly on a meridian
of longitude, it is 12
noon at all places
120°E
60°W
China
Canada (east)
along that meridian
Hong Kong
Barbados
(meridianmeans
‘midday’inLatin).
The Philippines
Trinidad
Malaysia
Venezuela
On the other side of
Western
Chile
the world along the
Australia
opposite meridian, it is
Midnight on the 60°W meridian
midnight. For example, Midday on the 120°E meridian
when it is noon in
Western Australia on the 120°E meridian, it is midnight in Chile on the 60°W meridian.
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Time
differences
SincetheEarthturns360°in24hours,itturns15°in1hour.Forevery15°oflongitude,the
timedifferenceis1hourandsoforevery1°oflongitudethetimedifferenceis4minutes.
SUMMARy
15° longitude = 1 hour time difference
1° longitude = 4 minutes time difference
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Local times around the world are measured relative to the time along the prime meridian 0°,
called Coordinated Universal Time (UTC) or Greenwich Mean Time (GMT).
Places east of the prime meridian are ahead of UTC, while places west are behind UTC.
UTC
–12h –11h –10h –9h –8h –7h –6h –5h –4h –3h –2h –1h
+1h +2h +3h +4h +5h +6h+ +7h 8h
0
Eastern hemisphere
London
Rome
Los Angeles
Athens
Cairo
New York
Jamaica
Moscow
Bangkok
Atlantic
Ocean
Rio de Janeiro
Indian
Ocean
Melbourne
180°W 175° 150° 135° 120° 105° 90° 75° 60° 45° 30° 15°
0°
International Date Line
Western hemisphere
+9h +10h +11h +12h
Pacific
Ocean
Fiji
Brisbane
New Zealand
15° 30° 45° 60° 75° 90° 105° 120° 135° 150° 165° 180°E
Example 9
The time in New York, USA is 5 hours behind UTC.
a What is the time in New York when it is 3 p.m. in London?
b What meridian of longitude passes through New York?
Solution
a 3 p.m. − 5 hours = 10 a.m.
b Difference in longitude = 5 × 15°
1 hour difference = 15° longitude.
= 75°
New York is behind UTC, so it is west of the prime meridian, with longitude 75°W.
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Example 10
In Los Angeles, USA (119°W), a tennis match is being played at 2:00 p.m. on Tuesday. Dave
is watching the match live on TV in Sydney (151°E). What is the day and time in Sydney?
Solution
UTC
Los Angeles
119°W
0°
Los
Angeles
151°E
Difference
in longitude
Sydney
119°W
0°
Sydney
151°E
Difference in longitude
Difference in longitude = 119° + 151° = 270°
270
Time difference =
= 18 hours
15
Sydney is east of Los Angeles, so Sydney time is ahead of Los Angeles time.
Sydney time = 2:00 p.m. Tuesday + 18 hours = 8:00 a.m. Wednesday
Example 11
When it is 9:00 p.m. in Mumbai, India (19°N, 73°E), what is the local time in Jakarta,
Indonesia (6°S, 106°E)?
Solution
73°E
0°
106°E
Difference in longitude = 106° − 73° = 33°
1
33
= 2 h = 2 h 12 min
15
5
(or Time difference = 33 × 4 min = 132 min = 2 h 12 min)
Time difference =
Jakarta is east of Mumbai, so its time is ahead of Mumbai’s.
Local time in Jakarta = 9:00 p.m. + 2 h 12 min = 11:12 p.m.
19°N
Mumbai
0°
6°S
Jakarta
In Example 11, the time in Jakarta will not really be 11:12 p.m. but will be 11:00 p.m.
because the world is divided into hourly time zones. However, for the following exercise we
shall ignore time zones and calculate ‘local times’, correct to the nearest minute.
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Exercise 9-03 Longitude and time differences
Example
9
1 The time in Tahiti, a Pacific island, is 10 hours behind UTC.
a What is the longitude of Tahiti?
b What is the time in London (UTC) when it is 6:30 a.m. in Tahiti?
c What is the time in Tahiti when it is 6:30 a.m. in London?
2 The time in Vienna, Austria (48°N, 16°E) is 3 hours behind the time in
Tehran, Iran.
a Calculate the difference in longitude between the two cities.
b Hence, what meridian of longitude goes through Tehran?
c When it is 11:15 p.m. in Tehran, what is the time in Vienna?
Example
10
3 Hanoi, Vietnam lies on the 105°E meridian while Sydney lies on the 150°E meridian.
When the time in Hanoi is 5:30 p.m., what is the time in:
a Sydney?
b London?
4 What is the local time in Pretoria, South Africa (25°S, 30°E) when it is 10:45 p.m. in
London? Select A, B, C or D.
A 8:45 p.m.
B 9:05 p.m.
C 12:25 a.m.
D 12:45 a.m.
5 What is the time in London when it is 7:10 p.m. in Rio de Janeiro, Brazil
(23°S, 43°W)?
6 When it is 2:30 p.m. in Alice Springs (134°E), what is UTC?
Example
11
7 Kalgoorlie, WA has coordinates (30°S, 121°E) while the Pacific island of Nauru has
coordinates (0°, 166°E).
a Calculate the difference in longitude between these two places.
b Calculate the time difference between the two places.
c What is the time in Nauru when it is 9:45 a.m. in Kalgoorlie?
8a
Calculate the difference in longitude between New Delhi, India (28°N, 77°E) and
Milan, Italy (41°N, 12°E).
b Calculate the time difference between New Delhi and Milan in hours and minutes.
c What is the local time in Milan when it is 5:00 p.m. in New Delhi?
9a
Calculate the time difference between Lisbon, Portugal (38°N, 9°W) and Taipei,
Taiwan (25°N, 121°E).
b What is the local time in Taipei when it is 11:20 a.m. in Lisbon?
10An ocean liner’s local time is 5:45 p.m. when UTC is 8:05 a.m. What is its longitude
position? Select A, B, C or D.
A35°E
B35°W
C145°E
D145°W
11Houston, USA has coordinates (29°N, 95°W) while Kabul, Afghanistan has
coordinates (34°N, 70°E). If it is 6:20 p.m. Sunday in Houston, what is the time and
day in Kabul?
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12Guatemala is 6 hours behind UTC.
a What meridian of longitude runs through Guatemala?
b What is the time in the UK when it is 1:30 p.m. in Guatemala?
13aWhen it is 7:30 p.m. Monday in Los Angeles, USA (34°N, 119°W), what is the time
and day in Sydney (34°S, 151°E)?
b A plane leaves Los Angeles at 7:30 p.m. Monday and flies 14 12 hours to get to Sydney.
What is the local time in Sydney when it arrives?
14When it is 7 a.m. in Broome, Western Australia, it is 11:40 p.m. the previous night, in
Munich, Germany.
a Calculate the difference in longitude between Broome and Munich.
b Munich has longitude 12°E. Calculate the longitude of Broome.
15When it is midday in Johannesburg, South Africa, it is 10:48 a.m. in Pisa, Italy.
a Calculate the difference in longitude between Johannesburg and Pisa.
b Pisa has longitude 10°E. Calculate the longitude of Johannesburg.
16When it is 9:30 p.m. in Ottawa, Canada, it is 3:02 a.m. the next day in Bern,
Switzerland.
a What is the time difference between the 2 cities?
b Calculate the difference in longitude between Ottawa and Bern.
c Bern has longitude 7°E. Calculate the longitude of Ottawa.
Just for the record
Daylight saving
In 1907, British builder William Willett noticed that the extra hours of daylight in the
summer mornings were being ‘wasted’ because people were still asleep at this time. He
proposed a scheme of putting clocks forward one hour so that people would wake up
earlier and have more daylight hours in the afternoon for recreation.
Daylight saving was first trialled in Britain and Australia during the First World War
as a means of conserving energy. In Australia, it was trialled again from 1971 and
was formally adopted in NSW in 1976 when people voted 13 to 6 in favour of it.
Queensland, Northern Territory and Western Australia voted against it and do not have
daylight saving.
a
How does daylight saving reduce crime, pedestrian fatalities and energy costs?
b
Daylight saving is less popular in country areas. Why do you think this is so?
c
Some countries have daylight saving in winter. Why do you think this is so?
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Table of time
zones
wS
Since1884,theworldhasbeendividedintostandardtimezones,withroughlyonehour
differenceforevery15°oflongitude.Somecountriesmakevariationsforgeographical
and/oradministrativereasons.Forexample,CentralAustralianTimeis9.5hours(not
9hours)aheadof
A B C D E F G H I J K L M N O P Q R S T U V W XY
UTC and China has
only one time zone
(UTC+8)eventhough
it stretches across
four. On the map
below, places along
the 105°E meridian
(columnS)are7
hours ahead of UTC,
while places along
the 150°W meridian
(columnB)are10
–12 –10 –8
–6
–4
–2
+2
+4
+6
+8 +10 +12
hours behind UTC.
UTC
International Date Line
Map of the
world
9-04 international time zones
International Date Line
wS
ThistableliststheinternationaltimezonesrelativetoUTCandAustralian Eastern
Standard time (AESt),thetimezoneforeasternAustralia,coveringQueensland,NSW,
ACT, Victoria and Tasmania.
Hours
from UTC
400
Hours from
AEST
Places in this zone
−11
−21
American Samoa, Niue Island, Midway Island
−10
−20
Hawaii, Cook Islands, Tahiti
−9
−19
Alaska
−8
−18
USA/Canada Pacific time (west coast), Los Angeles, Vancouver
−7
−17
USA/Canada Mountain time, Mexico (west)
−6
−16
USA/Canada Central time (mid-west), Mexico (east), El Salvador, Nicaragua,
Guatemala, Honduras, Belize
−5
−15
USA/Canada Eastern time (east coast), New York, Cuba, Peru
−4
−14
Canada Atlantic time, Chile, Barbados, Brazil (west), Bolivia
−3
−13
Argentina, Brazil (east), Uruguay, Greenland
−2
−12
South Sandwich Islands
−1
−11
Azores
0
−10
UK, Ireland, Iceland, Portugal, Ghana, Liberia, Mali, Morocco
+1
−9
Most of Europe, Algeria, Angola, Chad, Bosnia-Herzegovina, Croatia
+2
−8
Finland, Greece, Lebanon, Egypt, South Africa, Bulgaria, Cyprus
+3
−7
Russia (west), Saudi Arabia, Kenya, Madagascar, Iraq
+3.5
−6.5
+4
−6
Iran
Mauritius, United Arab Emirates, Armenia
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Hours
from UTC
Hours from
AEST
+4.5
−5.5
+5
−5
+5.5
−4.5
Places in this zone
Afghanistan
Pakistan, Maldives, Kazakhstan
India, Sri Lanka
+6
−4
+6.5
−3.5
+7
−3
Vietnam, Cambodia, Thailand, Indonesia (west), Laos
+8
−2
Australian Western Standard Time (WA), Malaysia, China, the Philippines,
Indonesia (central), Mongolia, Taiwan
+9
−1
+9.5
−0.5
+10
0
+10.5
+0.5
Bangladesh, Bhutan, Kazakhstan, Kyrgystan
Myanmar (Burma), Cocos Islands
Japan, North Korea, South Korea, Indonesia (east), East Timor
Australian Central Standard Time (NT, SA, Broken Hill)
Australian Eastern Standard Time, Papua New Guinea, Guam
Lord Howe Island
+11
+1
+11.5
+1.5
+12
+2
New Zealand, Fiji, Nauru
+13
+3
Samoa, Tonga
Vanuatu, New Caledonia, Solomon Islands, Micronesia
Norfolk Island
The International Date Line
The International Date Line (IDL), on the opposite side
of the world to the prime meridian, runs through the Pacific
Ocean between Fiji and Hawaii. It is shown as the red
dotted line on the map below, and is ‘bent’ to ensure that it
does not run through any countries.
N
EASTERN
HEMISPHERE
Hawaii
INTERNATIONAL
DATE LINE
10º N
EQUATOR
Kiribati
Fiji
Samoa
0º
10º S
Tahiti
West of the IDL is the UTC+12 time zone including New
Tonga
Australia
30º S
Zealand, Fiji and Nauru, while east of the IDL is the
New
UTC−11 time zone including American Samoa and Niue
Zealand
Pacific Ocean
180º
160ºE
160º W
140º W
Island. So if you cross the IDL travelling east, today becomes
WESTERN
HEMISPHERE
yesterday (for example, Wednesday becomes Tuesday)
and you experience the same day twice. The reverse occurs when you cross the IDL travelling
west: today becomes tomorrow (for example, Sunday becomes Monday) and you ‘lose’ a day.
Example 12
Habib is flying from New York, USA to Istanbul, Turkey to visit his grandfather. New
York is 5 hours behind UTC while Turkey is 2 hours ahead of UTC.
a Habib wants to call his grandfather on the phone before he leaves New York. At
what time should he call so that it is 6:00 p.m. in Turkey?
b Habib boards the plane at 1:45 p.m. New York time and the trip lasts 8.5 hours.
What is the local time in Istanbul when he arrives?
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Solution
New York
Istanbul
–5
0(UTC)
+2
a Time difference = 5 + 2 = 7 hours
New York is 7 hours behind Istanbul.
New York time = 6:00 p.m. − 7 hours = 11:00 a.m.
b Arrival time = 1:45 p.m. + 8.5 hours
New York time.
= 10:15 p.m.
= 10:15 p.m. + 7 hours
= 5:15 a.m. (next day)
Istanbul time.
Australian standard time zones
There are three time zones in Australia (ignoring daylight saving), shown on the map below.
Northern
Territory
Western
Australia
South
Australia
Queensland
New
South
Wales
Victoria
A.C.T.
Tasmaria
Australian Western
Standard time
(AWST) UTC+8
Australian
Australian Eastern
Central Standard Standard time
(AEST) UTC+10
time (ACST)
UTC+9.5
These time zones can also be shown on a timeline.
West
402
East
0
+1 12 h
+2 h
Western Standard Time
Perth
Central Standard Time
Adelaide
Alice Springs
Darwin
Broken Hill (NSW)
Eastern Standard Time
Sydney
Brisbane
Canberra
Hobart
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Example 13
a Pete is in Perth and it is 11:30 a.m. He wants to ring his mother in Sydney.
What time is it in Sydney?
b Kimberley has just finished work at 5 p.m. in Sydney. What time is it in
Adelaide?
Solution
a
0
Perth
+1 12 h
Adelaide
+2 h
Sydney
Moving from west (Perth) to east (Sydney) ADD 2 hours.
Time in Sydney = 11:30 a.m. + 2 hours = 1 p.m.
It is 1 p.m. in Sydney.
b Moving from east (Sydney) to central (Adelaide) SUBTRACT 12 hour or
30 minutes.
Time in Adelaide = 5 p.m. − 30 min = 4:30 p.m.
It is 4:30 p.m. in Adelaide.
Daylight saving
In summer, many countries adopt daylight saving time or summer time and turn their
clocks forward one hour to take advantage of the increased hours of sunlight. By ‘losing’ that
hour at the start of daylight saving, people are actually working and sleeping 1 hour earlier
and gaining maximum use of the daylight.
Daylight saving has been operating in Australia since 1971, although Queensland, Western
Australia and the Northern Territory do not participate in the scheme. Each year, it runs from
the first Sunday in October (mid-spring) to the last Sunday in April (mid-autumn).
summary
For daylight saving time, add one hour.
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This map shows the local times around Australia
during daylight saving when it is 12:00 midday
in NSW.
NT
10:30 am
QLD
11 am
WA
9 AM
SA
11:30 am
NSW/ACT
12 pm
VIC 12 pm
TAS
12 pm
Example 14
Simone, in Melbourne, wants to chat online with her cousin, Zac, in Vancouver, Canada.
Canadian Pacific Standard Time is 8 hours behind UTC while Melbourne is 10 hours
ahead of UTC. Daylight saving is operating in Canada. When should Simone logon to
reach Zac at 4 p.m. Friday in Vancouver?
Solution
Vancouver time is UTC−8 but adding one hour for daylight saving gives UTC−7.
+10 h
–7 h
Vancouver
Canada
0º
UTC
(Greenwich)
Melbourne
Australia
Time difference = 7 + 10 = 17 hours.
+17 h
Vancouver
4 pm
Melbourne
Melbourne is 17 hours ahead of Vancouver.
Logon time = 4:00 p.m. Friday + 17 hours
404
= 9:00 a.m. Saturday.
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Exercise 9-04 international time zones
Refer to the table on page 400 or the worksheet ‘Table of time zones’ for this exercise.
1
When it is 4:00 p.m. Saturday in Sydney, what is the time in:
a
2
4
c
Russia (west)?
d
Greece?
Tonga?
Greenland?
b
c
Bosnia-Herzegovina?
Example
14
d
Myanmar?
Table of time
zones
New Caledonia?
What is the time difference (in hours) between:
a
Malaysia and New Zealand?
b Brazil (east) and Chile?
c
Iceland and the Philippines?
d Algeria and Salt Lake City,
USA (Mountain time)?
The Australian Open Tennis competition is broadcast live from Melbourne at 8:30 p.m.
local time. At what time is it seen in:
a
5
b
When it is 10:00 a.m. Tuesday in London, UK, what is the time in:
a
3
India?
wS
b
Dallas, USA (Central time)?
c
Japan?
Example
13
Sweden (Europe)?
It takes 8 12 hours for a plane to fly from Sydney to Hawaii. If Heath leaves Sydney at
1:30 p.m. Friday, what is the local time when he arrives in Hawaii? Select A, B, C or D.
6
7
8
9
A 2:00 p.m. Thursday
B
5:30 p.m. Thursday
C
D
10:00 p.m. Friday
2:00 a.m. Friday
If the time in Sydney is 6:00 p.m., what is the time in:
a
Melbourne?
b
Hobart?
c
Alice Springs?
d
Perth?
e
Adelaide?
f
Surfers Paradise?
Queensland, Northern Territory and Western Australia do not participate in daylight
saving. If the daylight saving time in Sydney is 10:30 a.m., what is the time in:
a
Melbourne?
b
Hobart?
c
Alice Springs?
d
Perth?
e
Adelaide?
f
Surfers Paradise?
a
What is the time in Santiago, Chile when it is 12:00 noon in Perth?
b
What does this say about the positions of Perth and Santiago on the world globe?
A plane takes 12 hours to fly from Adelaide (UTC+9.5) to Cape Town, South Africa
(UTC+2). If Shaunna leaves Adelaide at 10:00 a.m., what is the local time when she
arrives in Cape Town?
10 Robert in Tasmania wants to call his brother in Scotland, UK when the time there is
4:30 p.m. daylight-saving time. At what time should Robert call? Select A, B, C or D.
A 1:30 a.m.
B
2:30 a.m.
C
3:30 a.m.
D
5:30 a.m.
11 Indira flew from New Delhi, India to Sydney, leaving at 6:30 p.m. and arriving the next
day at 12:00 p.m. local time. How long did her plane trip take?
12 The Superbowl is broadcast live from New York, USA (Eastern time) at 6:30 p.m.
Monday. At what time and day is it seen in:
a
Sydney?
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b
Berlin, Germany (Europe)?
9. Geometry of the Earth
c Dublin, Ireland?
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13Wallaby Airlines flies from Adelaide to Mt Isa, Queensland, via Brisbane. The flight left
Adelaide at 2:30 p.m. and arrived in Brisbane 2 h 40 min later.
The plane stayed in Brisbane for 35 minutes for refuelling and to take extra passengers
on board. The flight from Brisbane to Mt Isa took 2 h 45 min. Find the local time when
the plane arrived:
a in Brisbane
b in Mt Isa
14Laire boards a flight in Coffs Harbour at 9:50 a.m. Tuesday morning. The flight
to Sydney takes 40 minutes. Laire then waits in Sydney for one hour and boards a
flight to Perth, which takes 4 hours 25 minutes. What time is it in Perth when the
plane lands?
15A plane takes 2 hours and 45 minutes to fly from Wellington, New Zealand
(UTC+12) to Sydney (UTC+10). If Paula leaves Wellington at 4:20 p.m. (daylight
saving time) in September, what is the local time when she arrives in Sydney
(standard time)?
16A plane leaves Sydney at 1:00 p.m. Thursday and flies 20 hours to Paris, France. At what
time does the plane arrive in Paris if daylight saving is operating in France but not in
Sydney?
17A plane leaves Hawaii on Monday at 12 noon and flies 6 hours west to Tokyo, Japan.
a When the plane crosses the International Date Line, what day does it become?
b What is the local time in Tokyo when the plane arrives?
18Liz flew west from Buenos Aires, Argentina (UTC−3) to Melbourne (UTC+10). To her
surprise, she discovered that when she arrived in Melbourne it was the same time,
7:30 p.m., as when she left Buenos Aires.
a How long did her plane journey take?
Shutterstock.com/javarman
b Was it still the same day? If not, what day was it?
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investigation
Researching time zones
a Why is UTC also known as GMT, Z time, Zulu time or Zebra time?
b How is time measured at the North and South Poles, where all the meridians of
longitude meet?
c
How is time measured in Australia’s external territories, such as Norfolk Island,
Lord Howe Island, Cocos Islands and Christmas Island (Indian Ocean)?
d
Which country stretches across 11 time zones?
e
Australia, USA and Canada are roughly the same length, but Australia has three
time zones while USA and Canada have four and five time zones respectively. Why
do you think this is so? (Hint: Look at the countries on a world globe or map.)
investigation
Plan an overseas holiday
Use an atlas or a world globe to plan a trip to visit up to five different places in the world.
Design an itinerary that lists the places you’ll be visiting. You will need to take into
account:
• distances between places
• mode of transport and travelling speeds: by air, land or sea
• arrival and departure times
• international time zones
• climate/seasonal conditions (these also affect the popularity of destinations).
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Sample HSC problem
Lillian is travelling by plane from Cape Town, South Africa (34°S, 18°E) to Stockholm,
Sweden (59°N, 18°E) for a holiday.
a If the radius of the Earth is 6400 km, calculate, correct to the nearest kilometre, the
distance between Cape Town and Stockholm.
b If the plane flies at an average speed of 820 km/h, how long, in hours and minutes, would
a direct flight between Cape Town and Stockholm take?
c Lillian can only book a flight that takes her from Cape Town to Stockholm via London.
After boarding the plane at Cape Town at 9:45 a.m. on Tuesday, Lillian takes 12.5 hours
to fly to London, then after a 2-hour break she takes another 2 hours and 25 minutes to
fly to Stockholm. Calculate the local time and day Lillian arrives in Stockholm if the time
zones for Stockholm and Cape town are UTC+1 and UTC+2 respectively.
Study tip
Ten common errors made in HSC exams
1 Careless setting out of answers or ‘bald answers’ with no working
2 Careless errors in basic algebra
3 Not choosing the simplest and quickest method
4 Not reading or answering the whole question
5 Poorly drawn diagrams that are messy, small and without labels or detail
6 Not having a ruler: for drawing, measuring and graph reading
7 Rounding too early in a calculation, so that the final answer is inaccurate
8 Not checking whether a numerical answer makes sense or sounds reasonable
9 Difficulties in explaining answers in words: writing mini-essays rather than 1−2 sentences
that are to the point
10 Not realising that the parts of a question are related and require an answer or hint from
a previous part
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ChAptER REviEw
topic overview
wS
This short chapter, Geometry of the Earth, introduced entirely new concepts and skills related
to the geography and measurement of the world. Make sure that all theory, rules, terminology
andworkedexamplesareincludedinyoursummary.Thistopiccanbedividedintothree
neat sections: latitude and longitude, calculating distances and calculating times. Students
often experience difficulty in calculating differences in latitude, longitude and time, but such
confusioncanbeovercomebyachievingadeepandstructuredunderstandingofthetopic.
Crossword of
the Earth
Make a summary of this topic. Use the outline at the start of this chapter as a guide. An
incomplete mind map has been printed below. Use your own words, symbols, diagrams,
boxesandreminders.Gaina‘wholepicture’viewofthetopicandidentifyanyweakareas.
Great circle
distances
r = 6400 km
Latitude
and
longitude
GEOMETRY OF
THE EARTH
Longitude
and time
differences
Time
zones
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Revision
Exercise
9-01
1 What are the position coordinates of A, B, C and
D on the Earth?
70°W
Equator
O
30°
A
C
Exercise
9-01
B
D
55°S
2 Albury, NSW has coordinates (36°S, 147°E).
a What is the longitude of Albury?
b How many degrees is Albury north of the South Pole?
c Broome has coordinates (18°S, 122°E). What is the difference in longitude between
Albury and Broome?
Exercise
9-01
Exercise
9-02
Exercise
9-02
3 Amsterdam, The Netherlands is 15° north and 122° west of Seoul, South Korea (37°N,
127°E). What are the coordinates of Amsterdam?
4 Use the arc length formula to calculate the great circle distance, to the nearest kilometre,
between Tokyo, Japan (35°N, 139°E) and Mt Isa (20°S, 139°E) (radius of the Earth is
6400 km).
5a What great circle runs through S and T ?
b Calculate the shortest distance between S and T to the
nearest kilometre.
16°E
20°N
S
O
T
Exercise
9-03
40°S
6 The time in Baghdad, Iraq (33°N, 44°E) is 7 hours ahead of the Falkland Islands.
a What line of longitude runs through the Falkland Islands?
b When it is 3:40 p.m. in the Falkland Islands, what is the time in Baghdad?
Exercise
9-03
7a
Calculate the difference in longitude between Naples, Italy (41°N, 14°E) and Bali,
Indonesia (8°S, 115°E).
b Calculate the time difference between the two cities (ignoring time zones).
Exercise
9-04
c What is the local time and day in Bali when it is 5:30 p.m. Thursday in Naples?
8 The time in Anchorage, Alaska (USA) is 9 hours behind UTC while the time in Lusaka,
Zambia is 2 hours ahead of UTC.
a When the time in Anchorage is 8:00 a.m., what is the time in London?
b When the time in Lusaka is 6:30 p.m., what is the time in Anchorage?
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9 What is the time in Brisbane (AEST) when it is:
a 4 p.m. in Perth? b 6 a.m. in Adelaide? c 2:30 a.m. in Hobart?
10A plane leaves Darwin on Thursday at 6:00 p.m. and travels east for 8 hours to reach
Niue Island.
Exercise
9-04
Exercise
9-04
a When the plane crosses the International Date Line, what day does it become?
b If Darwin is UTC+9.5 and Niue Island is UTC−11, what is the time on Niue Island
when the plane arrives there?
11Kerrie, living in Perth, wants to call her sister in Los Angeles, USA when the time there
is 12 noon Sunday. At what time should she call if Perth is UTC+8 and Los Angeles is
usually UTC−8 but the USA is on daylight saving time?
ISBN 9780170238977
CLA-NCM12PW2-13-0104-009.indd 411
9. Geometry of the Earth
Exercise
9-04
411
11/07/13 11:43 AM