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Measurement 3:
Spherical Geometry
r Book - Series M-1
Teache
EXCEL HSC GENERAL
MATHEMATICS
pages 100, 102, 126
centre of a circle of radius
nearest cm.
n angle of 120° at O, the
A
B
72°
m
12 c
O
P
120°
15 cm
O
Q
Mathletics
Instant
Workbooks
rc PQ?
of a circle of radius 12 cm.
cm.
Copyright ©
of length 58 cm subtends
e of ∠AOB to the nearest
A
O
θ
58 cm
22
cm
Measurement
3: Spherical
Geometry
Measurement
– Spherical
geometry
Topic Test TOPIC TEST
Instructions
• 15 multiple-choice questions
Instructions
•
15
multiple
questions
• Each choice
question
is worth 1 mark
•
Each
question
is
worth
1 mark
• Fill in only ONE CIRCLE
• Fill in only ONE CIRCLE
• Calculators may be used
• Calculators may be used
Time allowed: 20 minutes
PART A
SECTION I
Total marks = 15
Time allowed: 20 minutes
Total marks: 15
1 Greenwich meridian is also called:
A Latitude
B
Longitude
C
meridian
D
prime meridian
C
10 knots
D
20 knots
C
180°
D
360°
2 One nautical mile per hour is equal to:
A 1 knot
B
2 knots
3 The longitude of the prime meridian is:
A 0°
B
90°
4 The radius of the Earth at the equator is approximately 6400 km. Find the circumference of the Earth at the
equator correct to the nearest km.
A 20 106 km
B
40 212 km
C
50 212 km
D
60 106 km
D
892 M
D
4M
D
33 knots
5 Given that 1 M = 1.852 km, 1560 km to the nearest nautical mile is:
A 800 M
B
842 M
C
862 M
6 Given that 1 M = 1.852 km, a distance of 3704 m is equal to:
A 1 M
B
2M
C
3M
7 Calculate the average speed, in knots, for a journey of 30 M in 3 hours.
A 90 knots
B
10 knots
C
20 knots
8 Find the distance to the nearest km between two points on the Earth’s surface that subtend an angle of 1°
at the centre. (Use the radius of the Earth = 6400 km.)
A 110 km
B
111 km
C
112 km
D
113 km
9 Adelaide is approximately (35° S, 139° E) while Tokyo is approximately at (35° N, 139° E). The distance
between Adelaide and Tokyo is approximately:
A 20 M
B
63 M
C
2400 M
D
4200 M
10 Find the time taken for a ship to sail between the two points A (60° N, 110° W) and B (20° S, 110° W) at an
average speed of 50 knots.
A 1 day
B
2 days
C
3 days
D
4 days
11 Ignoring time zones, the difference in time between two cities is 2.4 hours. In hours and minutes the time
difference is:
A 2 hours 4 minutes
B
2 hours 24 minutes
C 2 hours 40 minutes
D
none of these
45
CHAPTER 3 – Measurement – Spherical geometry
Measurement 3: Spherical geometry
Mathletics Instant Workbooks – Series M Copyright © 3P Learning
ii
12 City C has location (34° S, 18° E) and city D (15° N, 17° W). The difference in longitude is:
C 35°
A 19°
B 49°
D 1°
12 City C has location (34° S, 18° E) and city D (15° N, 17° W). The difference in longitude is:
13 P(37° N, 120° E) and Q(37° N, 120° W) are two towns. P and Q:
C 35°
A 19°
B 49°
D 1°
A both lie on the same great circle
B both lie on the same small circle
13 P(37° N, 120° E) and Q(37° N, 120° W) are two towns. P and Q:
C lie on neither the same great circle nor the same small circle
A both lie on the same great circle
B both lie on the same small circle
D there is insufficient information
C lie on neither the same great circle nor the same small circle
14 Two European cities, X and Y, are in different time zones. The difference in time between the cities is
1
insufficient
/2 there
hours.isCity
X is eastinformation
of city Y. If it is 10:35 a.m. in city X what is the time in city Y?
2D
14 Two
European
are p.m.
in different time
The difference inD time
p.m.cities, X and
8:05 a.m.
7:51between
a.m. the cities is
C zones.
A 12:51
B Y,1:05
21/2 hours. City X is east of city Y. If it is 10:35 a.m. in city X what is the time in city Y?
15 Two cities, L and M, have locations L(20° S, 70° W) and M(20° S, 47° E). Ignoring time zones, the difference
12:51
p.m. the two B
1:05
C 8:05 a.m.
D 7:51 a.m.
inA time
between
cities
is: p.m.
15 Two
and
M, have B
locations
L(20°
S, 70° W) and
M(20°
47°minutes
E). Ignoring
1 hour L32
minutes
2 hours
40 minutes
hoursS, 48
theyzones,
are on the
the difference
same time
C 7
A cities,
D time
in time between the two cities is:
B
2 hours 40 minutes C 7 Total
hours marks
48 minutes
are on the
D they
achieved
for PART
A same time
SECTION15 II
Instructions
• This section consists of 10 questions
TOPIC TEST
• Show all necessary working
Topic Test
SECTION II
PART B
Instructions • This section consists of 10 questions
Instructions
• 40
This minutes
section consists of 9 questions
Time
allowed:
• Show
all necessary working
Total marks: 37
• Show all necessary working
16
Time
allowed:45
40minutes
minutes
Time allowed:
2marks:
marks each
Total
Total
marks
= 4037
A
l
2 marks each
A
O 12 cm l B
60
°
a An arc makes an angle of 60° at the centre of a circle of radius 12 cm. Find
the length of the arc correct to the nearest centimetre.
16
a An arc makes an angle of 60° at the centre of a circle of radius 12 cm. Find
the length of the arc correct to the nearest centimetre.
60
°
A 1 hour 32 minutes
TOPIC TEST
O 12 cm
b In a circle of radius 20 cm, an arc PQ of length 54 cm subtends ∠POQ at
the centre. Find the size of ∠POQ to the nearest degree.
b In a circle of radius 20 cm, an arc PQ of length 54 cm subtends ∠POQ at
the centre. Find the size of ∠POQ to the nearest degree.
B
P
20 cm
P
O θ
54 cm
20 cm
O θ
Q
54 cm
c Two points on the surface of a sphere with radius 21 cm have an angular distance of 80° . Find their
Q
spherical distance.
c Two points on the surface of a sphere with radius 21 cm have an angular distance of 80° . Find their
spherical distance.
d Find the spherical distance between two points on the Earth’s surface if their angular distance is 65° .
(radius of the Earth = 6400 km)
d Find the spherical distance between two points on the Earth’s surface if their angular distance is 65° .
(radius of the Earth = 6400 km)
46
46
EXCEL HSC GENERAL MATHEMATICS REVISION & EXAM WORKBOOK
Measurement 3: Spherical geometry
EXCEL HSC GENERAL MATHEMATICS REVISION & EXAM WORKBOOK
Mathletics Instant Workbooks – Series M Copyright © 3P Learning
iii
17 Referring to the diagram at right, which point lies:
1 mark each
a on parallel of latitude 30° N
60°N
30° N
b on meridian of longitude 40° E
B
D
0°
c at (30° S, 40° W)
C
30° S
18 Referring to the diagram above, give the location of points: 1 mark each
A
F
E
60° S
a A
60° W
b B
60°E
40° W
19 For the points C and D in the above diagram, what is the difference in:
40° E
1 mark each
a latitude
b longitude
20 Given that 1.852 km = 1 M, find correct to 2 decimal places:
a
b
500 M in kilometres
2 marks each
750 km in nautical miles
21 Find the average speed in knots for journeys of:
a
84 M in 1 1 hours
2
2 marks each
b
2450 km in 15 hours
22 A, B, C and D are points on the Earth’s surface. A and B lie on parallel of latitude 30° N. C and D lie on the
equator. O is the centre of the Earth. ∠COD = 86° . A and C both have longitude 20° W. B and D have the
same longitude. (use 1.852 km = 1M)
a
1 mark
Find the longitude of B.
N
b
What is the distance from C to D in nautical miles?
1 mark
B
A
O
0°
86°
Find the distance from A to C in kilometres.
2 marks
D
C
20
c
°W
S
47
CHAPTER 3 – Measurement – Spherical geometry
Measurement 3: Spherical geometry
Mathletics Instant Workbooks – Series M Copyright © 3P Learning
iv
23 Ignoring time zones, find the difference in time between:
a
London (0° ) and Bali (115° E)
b
2 marks each
Sri Lanka (80° E) and Jamaica (77° W)
24 If it is 4:30 a.m. on 12th of May in Toronto, Canada (80° W), what is the time and date in Johannesburg,
South Africa (28° E).
3 marks
25 A plane leaves Sydney (150° E) at 8:20 p.m. and arrives in Dubai (55° E) at 6:20 a.m. local time. How long
was the flight?
3 marks
Total marks achieved for PART B
48
40
EXCEL HSC GENERAL MATHEMATICS REVISION & EXAM WORKBOOK
Measurement 3: Spherical geometry
Mathletics Instant Workbooks – Series M Copyright © 3P Learning
v
b 11 014 m2 19 74 m
74
4
46
D
60
C
46
90
Answers – Measurement 3:
Spherical geometry
A
Page 1
36 1 15 cm 2 a 94.25 cm b 1 c 31.42 cm 3 13 cm 4 151° 5 84 cm
3
2
Page 37
1 9494.6 km 2 6143.6 km 3 101° 52′ 4 147° 20′ 5 286.5 cm
Page 3
38 1 a G b M, H c L, B d D e X f J, P g E, K h A, F 2 a 15° N b 45° S c 60° N d 15° S
e 45° S f 75° N g 15° N h 60° S i 90° N j 90° S 3 a 20° b 60° c 60° d 20° e 120° f 100° g 80°
h 80° 4 a equator b 90° S, 90° N c latitude d equator
4
Page 39
1 a Greenwich b Greenwich c East, West d 0° 2 a D, F b A, H, I c G d B, C, L 3 a 30° W
b 125° E c 0° d 30° W e 0° f 60° E g 80° E h 125° E i 60° E j 125° E k 0° l 30° W 4 a 155° b 30° c 80°
d 80° e 155° f 155° g 65° h 90°
Page 40
1 a (75° N, 0° ) b (30° N, 0° ) c (50° S, 55° E) d (75° N, 55° E) e M f U 2 a great circle b small circle
5
c great circle d great circle e small circle f great circle 3 3000 M 4 a 5100 M b 3600 M
Page 6
41 1 13 963 km 2 11 170 km 3 a 52° E b i 4020 M ii 7484 km 4 a 7778 km b 8334 km
Page 42
1 a 741 km b 1204 km c 1296 km d 1806 km e 2334 km f 3704 km 2 a 189 M b 518 M
7
c 259 M d 2770 M e 1404 M f 4519 M 3 a 17.5 b 19.7 c 163.6 d 45 4 a 400 b 152.38 c 460
d 265.26 5 a 525.4 b 31.3
Page 43
1 a 3 h 56 min b 0 h c 1 h 52 min 2 Monday 3 3 p.m. 4 10:40 p.m. 5 9:52 a.m. on Wednesday 17 February
8
Page 9
44 1 a 10:40 p.m. on 27 June b 8:40 a.m. on 28 June 2 7:30 p.m. 3 4:56 a.m. on Tuesday 4 7:50 a.m. on 1st
of January
Pages 10–11
45–46 1 D 2 A 3 A 4 B 5 B 6 B 7 B 8 C 9 D 10 D 11 B 12 C 13 B 14 C 15 C
Pages 12–13
46–48 16 a 13 cm b 155° c 29.3 cm d 7261 km 17 a D b E c F 18 a (30° S, 60° W) b (60° N,
40° W) 19 a 30° b 120° 20 a 926.00 km b 404.97 M 21 a 56 knots b 88.2 knots 22 a 66° E b 5160 M
c 3334 km 23 a 7h 40 min b 10 h 28 min 24 11:42 a.m. on the 12th of May 25 16 h 20 min
H
H
Page 49 1 a
T
H
T
b 12
c
i
Page 50
2
3
1
ii
1
3
iii
3
5
a
2
5
d
68
95
e
27
95
f
51
190
T
1
2
1
2
R
1
3
2
4
B
R
1
4
B
H
HHH
T
HHT
H
T
H
HTH
HTT
THH
T
H
T
TH T
T TH
TTT
b i
3
10
R
B
b
1
8
c
3
8
d
7
8
e
1
8
2
2
a
3
4
ii
1
10
iii 3
5
2 a
5
11
b
3
11
c
6
11
3 a
12
13
14
21
23
24
31
32
34
41
42
43
2
3
4
1
3
4
1
2
4
1
2
3
1
3
20
b
51
380
3
190
c
5
b 5
c 1 d 5
e 5 2 4
3 0.08
18
18
6
18
9
25
1 a 24 b 120 c 720 2 mn 3 8 4 9 5 343 6 17 576 000 7 a 100 000 000 b 90 000 000
1 a 20 b 210 c 360 d 120 2 a 72 b 504 c 60 480 3 2520 4 15 600 5 990 6 9
Page 51 1 a
Page 52
Page 53
199
ANSWERS
Measurement 3: Spherical geometry
Mathletics Instant Workbooks – Series M Copyright © 3P Learning
vi