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# Maths A - Chapter 6

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### Maths A - Chapter 6

1. 1. 6syllabussyllabusrrefefererenceence Strand: Applied geometry Core topic: Elements of applied geometry In thisIn this chachapterpter 6A Latitude and longitude 6B Distances on the Earth’s surface 6C Time zones Earth geometry MQ Maths A Yr 11 - 06 Page 207 Wednesday, July 4, 2001 5:39 PM
2. 2. 208 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Introduction The early sailors and land explorers who roamed the world did not have sophisticated navigating devices. The sun, moon and stars served as their basic guide to local time. To determine their position on Earth, they required a reference point. The point chosen was Greenwich, England, through which the Prime Meridian passes, and the line where east meets west. In this chapter we shall explore questions such as the following. 1. Why is Greenwich so important in our daily lives? 2. Since the Earth approximates the shape of a sphere, the distance between two points on its surface is not represented by the length of a straight line, but by the length of an arc of a circle. How is this distance measured? 3. How do we measure time? What is Greenwich Mean Time? 4. At any particular instant, how can we compare the times at different positions on the Earth? Let us begin by researching the historical signiﬁcance of Greenwich and the Prime Meridian. The Prime Meridian Resources: Searching via the World Wide Web. A great deal of information about the Prime Meridian (or Greenwich Meridian) and related topics is available on the Web. This investigation is designed to enhance your research skills in retrieving relevant information using the World Wide Web. As a starting point, you may wish to explore the following web sites, then discover others for yourself. 1. http://greenwichengland.com/ 2. http://greenwich2000.com/ 3. http://greenwichmeridian.com/ 4. http://ordsvy.gov.uk/ 1 Systematically search the Web to answer the following questions. a Just where in England is Greenwich? b Why is Greenwich the centre of time and space? c What is meant by the term Prime Meridian? d Why does the Prime Meridian pass through Greenwich? What difference would it make if the Prime Meridian passed through another place on Earth, for example Brisbane? e There have been many ‘Greenwich Meridians’. How did this occur? f Why is the date October 1884 so important? 2 After gathering information from the Web (and other sources if necessary), write up your report as a practice research assignment.Your presentation should indicate evidence of detailed research into: a the difﬁculties faced by the early explorers in navigating around the globe b the signiﬁcance of Greenwich c the importance of the Prime Meridian and its history d the effect on the whole world of choosing another position for the Prime Meridian. This research could form the basis of a more formal assessment item. inv estigat ioninv estigat ion MQ Maths A Yr 11 - 06 Page 208 Wednesday, July 4, 2001 5:39 PM
3. 3. C h a p t e r 6 E a r t h g e o m e t r y 209 1 Lines of latitude and longitude are imaginary lines which circle the Earth. Explain the difference between the two. 2 Draw a sphere to represent the shape of the Earth. a Mark the equator, Tropic of Cancer, Tropic of Capricorn, North and South Poles. b Give the position of each of these lines of latitude in degrees. 3 Draw another sphere. On it mark lines representing: a 0° longitude b 20°E longitude c 50°W longitude. 4 Draw one more sphere. On it mark: a the radius and circumference of the equator b the radius and circumference of the Tropic of Capricorn. 5 The surface of the Earth approximates the shape of a sphere. We would normally consider the shortest distance between two points to be a straight line. Explain why this is not the case when measuring the distance between two points on the Earth. 6 If the radius of the equator is 6371 km, what is the length of the equator? 7 Eplain the following: a GMT b International Date Line c EST. 8 How many time zones exist within Australia? What are they? 9 What is daylight summer time? 10 Explain EST with respect to GMT. Position on Earth As previously mentioned, the shape of the Earth approximates a sphere. This means that, in measuring the distance between two places on its surface, we are really meas- uring the distance along the arc of a circle. To measure the distance around the equator, we are actually ﬁnding the length of the circumference of a circle. To calculate distances between points on the Earth’s surface, we must ﬁrst consider the location of points. Great circles and small circles Consider the sphere drawn on the right. The axis of the sphere is a diameter of that sphere. The ends of the axis are called the poles. If we draw any direct line around the sphere passing through both poles, a great circle is formed. A great circle is the largest possible circle that can be drawn around the sphere. Pole Axis Pole MQ Maths A Yr 11 - 06 Page 209 Wednesday, July 4, 2001 5:39 PM
4. 4. 210 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d The length of a great circle is found using the formulas for the circumference of a circle: C = πD, where D is the diameter of the sphere or C = 2πr, where r is the radius of the sphere. Now consider a circle drawn perpendicular to the axis of the sphere. Only one circle, called the equator, will be a great circle. The centre of the equator will be the centre of the sphere as shown on the right. Other circles that are perpendicular to the axis of the sphere will be smaller than a great circle and are called small circles. To calculate the length around a small circle, we need to know the small circle’s radius. The small circle will have a radius smaller than that of the great circle, as shown in the ﬁgure on the right. Great circle
5. 5. C h a p t e r 6 E a r t h g e o m e t r y 211 Latitude and longitude As the Earth is a sphere, great circles and small circles on the surface of the Earth are used to locate points on the surface. Consider the axis of the Earth to be the diameter joining the North Pole and the South Pole. The only great circle that is perpendicular to this axis is the equator. The angular distance either north or south of the equator is the latitude. Small circles parallel to the equator are called paral- lels of latitude. These small circles are used to describe how far north or south of the equator a place is located. For example, Brisbane lies close to the small circle 27°S. This means that Brisbane subtends a 27° angle at the centre of the Earth and is south of the equator. The maximum latitude for any point on the Earth is 90°N or 90°S. The North and South Poles lie at these points. For latitude, the equator is the line of reference for all measurements. To locate a place on the globe in an east–west direction, the line of reference is the Greenwich Meridian. The Greenwich Meridian is half a great circle running from the North to the South Pole. The Greenwich Meridian is named after Greenwich, a suburb of London. All other places on the globe can be described as being east or west of the Greenwich Meridian. The half great circle on which a place lies is called a mer- idian of longitude. Each meridian of longitude is identiﬁed by the angle between it and the Greenwich Meridian and by whether it is east or west of Greenwich. The meridian of longitude opposite the Greenwich Meridian is the International Date Line. The Inter- national Date Line has longitude 180° either east or west. On either side of the International Date Line the day changes. (This will be explained in more detail later in the chapter.) For the convenience of some small island nations and Russia, the International Date Line is bent so as not to pass through them. World maps or globes are drawn with both parallels of latitude and meridians of longitude shown and can then be used to locate cities. Each point can then be given a pair of coordinates. First, the parallel of lati- tude it lies on, followed by the meridian of longitude. For example, the coordinates of Brisbane are 27°S, 153°E. Equator South Pole North Pole 27°S 27° Brisbane GreenwichMeridian Equator South Pole North Pole 40° International Date Line Greenwich Meridian MQ Maths A Yr 11 - 06 Page 211 Wednesday, July 4, 2001 5:39 PM
6. 6. RUSSIA AUSTRALIA Sydney Brisbane Darwin LosAngeles VancouverMontreal NewYork RiodeJaneiro Perth Melbourne HobartAuckland PortMoresby NEW ZEALAND JAPAN NORTHKOREA SOUTHKOREACHINA INDIA SINGAPORE IRAQ SRILANKA TAIWAN PHILIPPINES INDONESIAPAPUA NEWGUINEA Moscow FINLAND SWEDEN 60ºW 60ºN 30ºN 0º 30ºS 60ºN 30ºN 0º 30ºS 30ºW150ºW120ºW90ºW60ºW30ºW0º0º30ºE60ºE90ºE120ºE150ºE180º 30ºW0º150ºW120ºW90ºW60ºW30ºW30ºE60ºE90ºE120ºE150ºE180º NORWAY ICELAND GREENLAND IRELAND UK ITALYFRANCE SPAIN ALGERIA MALI NIGER ANGOLA KENYA NAMIBIA SOUTHAFRICA MADAGASCAR Johannesburg LIBYA EGYPT PORTUGAL NETHERLANDS UNITEDSTATES OFAMERICA CANADA Alaska (USA) InternationalDateLine MEXICO Hawaii(USA) COLOMBIA JAMAICA ECUADOR PERU CHILE BOLIVIA PARAGUAY URUGUAY BRAZIL ARGENTINA BuenosAires Santiago Lima Colombo Oslo Suva FIJI CapeTown Manila ShanghaiBaghdad Tokyo Beijing Cairo Amsterdam Rome London Madrid 212 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d MQ Maths A Yr 11 - 06 Page 212 Monday, September 24, 2001 7:15 AM
7. 7. C h a p t e r 6 E a r t h g e o m e t r y 213 Identify the major cities closest to each of the following locations using the map on page 212. a 30°S, 30°E b 30°N, 120°E c 45°N, 75°W THINK WRITE a Look for the city closest to the intersection of the 30°S parallel of latitude and the 30°E meridian of longitude. a Johannesburg b Look for the city closest to the intersection of the 30°N parallel of latitude and the 120°E meridian of longitude. b Shanghai c Look for the city closest to the intersection of the 45°N parallel of latitude and the 75°W meridian of longitude. c Montreal 3WORKEDExample Write down the approximate coordinates of each of the following cities using the map on page 212. a Singapore b Perth c Los Angeles THINK WRITE a Use the parallels of latitude drawn to estimate the latitude. a Use the meridians of longitude drawn to estimate the longitude. 1°N, 104°E b Use the parallels of latitude drawn to estimate the latitude. b Use the meridians of longitude drawn to estimate the longitude. 32°S, 115°E c Use the parallels of latitude drawn to estimate the latitude. c Use the meridians of longitude drawn to estimate the longitude. 35°N, 118°W 1 2 1 2 1 2 4WORKEDExample MQ Maths A Yr 11 - 06 Page 213 Wednesday, July 4, 2001 5:39 PM
9. 9. C h a p t e r 6 E a r t h g e o m e t r y 215 Latitude and longitude For the following questions use the map on page 212. 1 Write down the name of the city closest to each of the following pairs of coordinates. a 30°N, 30°E b 30°N, 120°E c 15°S, 135°E d 45°N, 75°W e 50°N, 0° f 37°S, 175°E g 35°N, 140°E h 40°N, 115°E i 22 °S, 43°W j 60°N, 11°E 2 State the approxi- mate latitude and longitude of each of the following major cities or islands. a Melbourne b New York c Jamaica d Johannesburg e Rome f Buenos Aires g Baghdad h Moscow i Singapore j Suva Important parallels of latitude Four signiﬁcant parallels of latitude on the surface of the Earth are the: 1. Arctic Circle 2. Antarctic Circle 3. Tropic of Cancer 4. Tropic of Capricorn. Find out the latitude of these small circles and state the signiﬁcance of each. 6A WORKED Example 3 1 2 --- WORKED Example 4 Work SHEET 6.1 inv estigat ioninv estigat ion MQ Maths A Yr 11 - 06 Page 215 Wednesday, July 4, 2001 5:39 PM
10. 10. 216 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Distances on the Earth’s surface In this section, we shall consider distances between points on the same great circle, or the same small circle. Points on the same great circle All meridians of longitude are half great circles. The equator is also a great circle. This means that measuring the distance between two points on the same meridian of longitude, or between two points on the equator, involves calculating the length of an arc of a great circle. Consider a meridian of longitude on the Earth’s surface with two points on it. The angular distance between them will be the difference between their latitudes. The angular distance is calculated by subtracting the latitudes of points if both are on the same side of the equator and adding the latitudes if on opposite sides of the equator. The radius of the Earth is 6371 km. Calculating the circumference of a great circle on the Earth, we ﬁnd: C = 2πr = 2 × π × 6371 km = 40 030 km So the distance around a great circle on the Earth is 40 030 km. This also represents an angular distance of 360º. So 360º is equivalent to a distance of 40 030 km; that is: 360º ≡ 40 030 km therefore 1º ≡ 111.2 km Therefore, we have calculated that an angular distance of 1º on a great circle represents a distance of 111.2 km on the surface of the Earth. θ The coordinates of A are (20°S, 130°E) and the coordinates of B are (15°N, 130°E). Find the angular distance between them. THINK WRITE A and B are on opposite sides of the equator on the same meridian (130°E), so add the latitudes. Angular distance = 20° + 15° = 35° 5WORKEDExample 6371 km MQ Maths A Yr 11 - 06 Page 216 Wednesday, July 4, 2001 5:39 PM
11. 11. C h a p t e r 6 E a r t h g e o m e t r y 217 Points on the same small circle Points on the same parallel of latitude (other than the equator) lie on the circumference of a small circle. To ﬁnd the distance between two such points, we need to calculate the length of the arc of the small circle. This means that we need to know the radius of the small circle. Consider the diagram at right, showing a parallel of latitude at an angular distance θ from the equator. (This could be a parallel of latitude north or south of the equator.) To calculate the radius (r) of this small circle, we need to apply trigonometry. Angle ABC is equal to θ because of alternate angles on parallel lines. (The equator’s radius is parallel to the radius of the parallel of latitude.) The radius of the Earth is 6371 km, so: cos θ = = ∴ r = 6371 cos θ To calculate the circumference of the small circle: C = 2πr C = 2 × π × 6371 cos θ km = 40 030 cos θ km The circumference of the circle represents an angular distance of 360°, so: 360° ≡ 40 030 cos θ km ∴ 1° ≡ 111.2 cos θ km It is not necessary to reproduce the previous working in calculating distances between points on the same parallel of latitude. If we ﬁnd the angular distance between the two points, we can then apply the relationship that every degree of separation on a small circle is equivalent to a distance of 111.2 cos θ km where θ is the angular distance of the parallel of latitude from the equator. P and Q are two points on the Earth’s surface with coordinates (27°N, 30°W) and (39°N, 30°W) respectively. Calculate the distance between P and Q (to the nearest kilometre). THINK WRITE P and Q are on the same great circle. Calculate the angular distance, PQ. Angular distance = 39° − 27° = 12° Convert the angular distance to kilometres using 1° = 111.2 km. Distance = 12 × 111.2 km = 1334 km 1 2 3 6WORKEDExample C A r B θ rA C B 6371 km θ SkillS HEET 6.1 SkillS HEET 6.2 adjacent hypotenuse --------------------------- r 6371 ------------ MQ Maths A Yr 11 - 06 Page 217 Wednesday, July 4, 2001 5:39 PM
12. 12. 218 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d On a small circle: 1° ≡ 111.2 cos θ km where θ = degree of latitude. We must bear in mind that, in determining the distance between two points on the Earth’s surface, we require the shortest distance. This means that if the angular distance between two points is greater than 180º, a shorter distance would be obtained by measuring the distance on the minor arc, rather than the major arc. Arc length Find the distance (to the nearest km) between two places whose positions are (40°N, 170°W) and (40°N, 26°E). THINK WRITE These points are on the same parallel of latitude (40°N) so both lie on a small circle. Calculate the angular distance between points. They are on opposite sides of the Greenwich Meridian; so add degrees. Angular distance = 170° + 26° Angular distance = 196° This angular distance is >180°, so the shorter distance is required. Shortest angular distance = 360° − 196° Shortest angular distance = 164° Use the conversion equation for distance on small circles: 1° ≡ 111.2 cos θ Distance = 164 × 111.2 cos 40° Distance = 13 970 km 1 2 3 4 7WORKEDExample remember For points on the same great circle 1. Angular distance can be found by: (a) subtracting the latitudes if the points are on the same side of the equator (b) adding the latitudes if the points are on opposite sides of the equator. 2. The angular distance can be converted to a linear distance using 1° ≡ 111.2 km For points on the same small circle 1. Angular distance can be found by: (a) subtracting the longitudes if the points are on the same side of the Greenwich Meridian (b) adding the longitudes if the points are on opposite sides of the Greenwich Meridian. 2. The angular distance can be converted to a linear distance using 1° ≡ 111.2 cos θ km where θ = degree of latitude. For points on the equator or the same small circle Remember, if the angular distance is >180°, the shortest distance involves using the minor arc, which is found by subtracting the angular distance from 360°. remember MQ Maths A Yr 11 - 06 Page 218 Wednesday, July 4, 2001 5:39 PM
13. 13. C h a p t e r 6 E a r t h g e o m e t r y 219 Distances on the Earth’s surface 1 Two points, A and B, on the Earth’s surface are at (30°N, 25°W) and (20°S, 25°W). Calculate the angular distance between A and B. 2 In each of the following calculate the angular distance between the pairs of points given. a (70°N, 150°E) and (30°N, 150°E) b (25°N, 40°W) and (15°S, 40°W) c (64°N, 0°) and (7°S, 0°) d (42°S, 97°W) and (21°S, 97°W) e (0°, 60°E) and (0°, 20°W) 3 The city of Durban is at approximately (30°S, 30°E) while Cairo is at (30°N, 30°E). What is the angular distance between Durban and Cairo? 4 P and Q are two points on the Earth’s surface with coordinates (45°N, 10°W) and (15°N, 10°W) respectively. Calculate the distance between P and Q to the nearest kilometre. 5 Calculate the distance between each of the points below to the nearest kilometre. a A(10°N, 45°E) and B(25°S, 45°E) b C(75°N, 86°W) and D(60°S, 86°W) c E(46°S, 52°W) and F(7°S, 52°W) d G(34°N, 172°E) and H(62°S, 172°E) 6 The city of Osaka is at (37°N, 135°E) while Alice Springs is at (23°S, 135°E). Calculate the distance between Osaka and Alice Springs to the nearest kilometre. 7 The Tropic of Cancer is at latitude 23 °N while the Tropic of Capricorn is at latitude 23 °S. Calculate the distance between these two tropics along the same great circle in kilometres (correct to the nearest km). 8 M and N are two points on the Earth’s surface with coordinates (56°N, 122°W) and (3°S, 122°W). Calculate the distance, MN, correct to the nearest 100 km. 9 Calculate the distance between each of the points below, correct to the nearest kilometre. a P(85°S, 89°E) and Q(46°S, 89°E) b R(24°N, 0°) and S(12°S, 0°) c T(34°S, 17°W) and U(0°, 17°W) 10 Perth is at approximately (31°S, 115°E) while Hong Kong is approximately at (22°N, 115°E). The distance between Perth and Hong Kong is approximately: A 19 km B 98 km C 1000 km D 5890 km 11 Rachel is a ﬂight navigator. She is responsible for calculating the distance between Stock- holm (60°N, 18°E) and Budapest (47°N, 18°E). Rachel’s answer would be closest to: A 1446 km B 1452 km C 11 952 km D 11 890 km 12 Quito (0°, 78°W) and Kampala (0°, 32°E) are two cities on the equator. a Calculate the angular distance between Quito and Kampala. b Calculate the distance between them to the nearest 100 kilometres. 13 Calculate the distance between the North Pole and the South Pole in kilometres. 14 The city of Kingston is approximately at (18°N, 76°W). Ottawa is at approximately (46°N, 76°W). a Calculate the angular distance between Kingston and Ottawa. b Calculate the distance between Kingston and Ottawa to the nearest kilometre. 6B WORKED Example 5 WORKED Example 6 1 2 --- 1 2 --- mmultiple choiceultiple choice mmultiple choiceultiple choice MQ Maths A Yr 11 - 06 Page 219 Wednesday, July 4, 2001 5:39 PM
14. 14. 220 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 15 Find the distance (in km) between two places on the equator which have a difference in their longitudes of 35°. 16 Two places on the equator have an angular distance of 200°. What is the shortest dis- tance between these two places (to the nearest kilometre). 17 Find the distance in kilometres from each of the following places to the closer pole. Indicate which pole (North or South) is the closer. a Warwick (28°S, 152°E) b Vancouver (49°N, 123°W) c St Moritz (46°N, 10°E) d Thursday Island (10°S, 142°E) 18 Find the shortest distance (in km) between the following places. a 40°S, 130°E and 40°S, 159°E b 70°N, 15°E and 70°N, 100°E c 50°S, 66°W and 50°S, 106°W d 80°S, 67°W and 80°S, 89°W e 20°S, 150°E and 20°S, 54°W f 30°N, 28°E and 30°N, 39°W Variation of distance between points on given lines of longitude The aim of this investigation is to discover how the distance between two points on speciﬁed lines of longitude and the same line of latitude varies on progressing from the equator to the pole. Consider two lines of longitude, 0º and 100ºE with point P1 on 0º and point P2 on 100ºE, both on 0º latitude; that is, the equator. As these points move from the equator to the pole on the same line of latitude we are going to investigate the distance separating the two points. 1 Copy and complete the following table. For point P1 on 0º longitude and point P2 on 100ºE: 2 Describe what happens to the distance between P1 and P2 as we move from the equator to the pole. Is there a constant change? 3 You would perhaps assume that, at a latitude of 45º, the distance between P1 and P2 is half the distance between the points at the equator. This is not the case. Investigate to ﬁnd the line of latitude where this occurs (to the nearest degree). WORKED Example 7 inv estigat ioninv estigat ion Latitude Distance between P1 and P2 0º 10º 20º 30º 40º 50º 60º 70º 80º 90º MQ Maths A Yr 11 - 06 Page 220 Wednesday, July 4, 2001 5:39 PM
15. 15. C h a p t e r 6 E a r t h g e o m e t r y 221 Time zones As the Earth rotates, different parts of the globe are experiencing day and night at the same instant. This means that each meridian of longitude on the Earth’s sur- face should have a different time of day. To simplify this, the Earth is divided into time zones. Time on Earth Resources: Searching via the World Wide Web and a telephone directory. Our Earth takes 24 hours to complete one revolution on its axis. During that time, it rotates through 360º, and each place on earth experiences a period of daylight and a period of night time (except for places close to the poles where extended periods of day and night occur). So a 360º movement around the globe takes 24 hours; that is, a 1º movement around the globe takes hour. This means that every degree difference east or west on the globe is equivalent to a time difference of hour or 4 minutes. It would be an advantage to print out some of the time zone maps available on the Web for reference later. 1 This investigation requires research via the World Wide Web to provide answers to questions such as: a How did the early explorers determine local time? b What is a chronometer? c What is ZULU time? d What is the signiﬁcance of GMT? e What are ‘time zones’? Why are they necessary? f What are the time zones in Australia? What is the local time where you live (with respect to GMT)? g How are times throughout the world determined? inv estigat ioninv estigat ion 24 360 --------- 1 15 ------ MQ Maths A Yr 11 - 06 Page 221 Wednesday, July 4, 2001 5:39 PM
16. 16. 222 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d h Would local times on our globe be affected if the Prime Meridian were relocated to another position; for example, through your town? Explain in detail. i What is the resulting effect when a state or country anywhere throughout the world implements Daylight Saving Time? 2 To begin your search, you may wish to consult the following Web sites, then pursue some of your own. a http://www.rog.nmm.ac.uk/ b http://greenwichmeantime.com/ c http://www.worldtimezone.com/ 3 Most telephone directories contain a section displaying local times (with respect to GMT) for towns and cities around the world. Consult your local telephone directory. Compare the information displayed there with that obtained from the Web. 4 Organise your information in the form of a report or presentation to the class. Australian time zones Resources: World Wide Web, atlas, telephone directory. During periods when daylight saving time does not apply, Australia is divided into three time zones: Eastern Standard Time (EST), Central Standard Time (CST) and Western Standard Time (WST). 1 Which States are in each of the three time zones? 2 What is the time difference between each of these zones? 3 In which States does daylight saving time apply in summer? 4 When daylight saving is in force in the States that have it, how many time zones does Australia have? inv estigat ioninv estigat ion Australia Time Zone Map Daylight Saving Time Darwin Perth Adelaide Brisbane Sydney Canberra Melbourne Hobart Western Australia South Australia Northern Territory Queensland New South Wales Victoria Tasmania 01:44P 13:44 03:14P 04:14P 16:14 15:14 03:44P 15:44 04:44P 16:44 16:44 16:44 11-13-2000 Map outline © WorldTimeZone.com Compiled by Alexander Krivenyshev MQ Maths A Yr 11 - 06 Page 222 Wednesday, July 4, 2001 5:39 PM