Lesson Notes By Weeks and Term v3 - Senior Secondary 3
Longitude and latitude
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Longitude and latitude
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Subject: General Mathematics
Class: Senior Secondary 3
Term: 1st Term
Week: 3
Theme: Geometry
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Describe the earth as as phere Identify (using skeletalglobe O and locate (usingreal globe) and appreciate the. followings- North and Southpoles- Longitude- Latitude Meridian and -equator- Parallel of latitude Radius of parallel of -latitude- Radius of the earth. Recall, state for mulaand solve problems on:arc length of a curve Solve problems on longitude and latitude
For mathematical calculations in Senior Secondary 3, the Earth is approximated as a perfect sphere. Although its true shape is an oblate spheroid (slightly flattened at the poles and bulging at the Equator), treating it as a sphere simplifies calculations without significant loss of accuracy at this level.
Centre of the Earth (O): The imaginary point at the heart of the spherical Earth.
Radius of the Earth (R): The distance from the centre of the Earth to any point on its surface. Standard values often used are 6370 km or 6400 km.
North Pole (N): The northernmost point on the Earth's surface, where the Earth's axis of rotation emerges in the Northern Hemisphere.
South Pole (S): The southernmost point on the Earth's surface, where the Earth's axis of rotation emerges in the Southern Hemisphere.
Equator: This is a Great Circle (defined below) on the Earth's surface, equidistant from the North and South Poles. It divides the Earth into the Northern Hemisphere and the Southern Hemisphere. Its latitude is 0°.
Latitude: Definition: The angular distance of a point North or South of the Equator, measured along a meridian.
Parallels of Latitude: Imaginary circles drawn parallel to the Equator. All points on a given parallel of latitude have the same latitude. Latitudes range from 0° at the Equator to 90°N at the North Pole and 90°S at the South Pole.
Longitude: Definition: The angular distance of a point East or West of the Prime Meridian, measured along a parallel of latitude.
Meridians (Lines of Longitude): Imaginary semi-great circles that run from the North Pole to the South Pole, intersecting the Equator at right angles.
Prime Meridian (Greenwich Meridian): The meridian designated as 0° longitude. It passes through Greenwich, London. Longitudes range from 0° to 180°E (East) and 0° to 180°W (West). The 180° meridian is also known as the International Date Line.
Great Circle: A circle on the surface of a sphere whose plane passes through the centre of the sphere. The Equator is a great circle. All meridians of longitude are semi-great circles (halves of great circles). The shortest distance between any two points on the Earth's surface lies along the arc of a great circle passing through them.
Small Circle: A circle on the surface of a sphere whose plane does not pass through the centre of the sphere. All parallels of latitude (except the Equator) are small circles. For any parallel of latitude, say at an angle θ (theta) North or South of the Equator, its radius 'r' is related to the Earth's radius 'R' by the formula: `r = R cos θ` Where θ is the latitude of the parallel.
Illustration: Imagine a cross-section of the Earth through the poles. The Equator is a line across the centre. A parallel of latitude is a horizontal line above or below the Equator. The angle θ is formed between the radius of the Earth to a point on the parallel and the Equator. The radius 'r' of the parallel of latitude is the horizontal distance from the Earth's axis to the point on the parallel.
Example 1: Identifying Features and Radius of Parallel A point P is located at 60°N, 10°E. Assume the Earth's radius R = 6370 km. (a) Describe the Earth's approximate shape for mathematical purposes. (b) Identify the great circle that passes through P. (c) Calculate the radius of the parallel of latitude on which P lies.
Solution 1: (a) The Earth is approximated as a sphere for mathematical purposes in this context. (b) The great circle that passes through P (60°N, 10°E) is the meridian of longitude 10°E. (
Note: The Equator is also a great circle, but it does not pass through P, which is at 60°N). (c) The parallel of latitude is 60°
N. Using the formula `r = R cos θ`: `r = 6370 * cos(60°)` `r = 6370 * 0.5` `r = 3185 km` The radius of the parallel of latitude 60°N is 3185 km.
Example 2: Distance along a Great Circle (Same Longitude) A ship sails from Port Harcourt (4°N, 7°E) to Calabar (5°N, 7°E). Assuming the Earth's radius R = 6370 km and π = 22/7, calculate the distance covered by the ship along their common meridian.
Solution 2: Identify the coordinates: Port Harcourt (4°N, 7°E), Calabar (5°N, 7°E).
Determine path type: The longitude is the same (7°E), so the path is along a great circle (a meridian). Calculate angular difference (α): Both points are in the Northern Hemisphere (N). `α = |5°N - 4°N| = 1°`.
Apply great circle distance formula: `Distance = (α / 360°) * 2πR` `Distance = (1 / 360) 2 (22/7) * 6370` `Distance = (1 / 360) 44 910` `Distance = 40040 / 360` `Distance ≈ 111.22 km` The distance covered by the ship is approximately 111.22 km.
Example 3: Distance along a Parallel of Latitude (Same Latitude) An aircraft flies from Abuja (9°N, 7°E) to Ndjamena, Chad (9°N, 15°E) along the parallel of latitude 9°
N. Given R = 6370 km and π = 22/7, calculate the distance flown.
Solution 3: Identify the coordinates: Abuja (9°N, 7°E), Ndjamena (9°N, 15°E).
Determine path type: The latitude is the same (9°N), so the path is along a small circle (a parallel of latitude). Calculate angular difference (α): Both longitudes are East (E). `α = |15°E - 7°E| = 8°`.
Calculate radius of parallel (r): `r = R cos θ = 6370 cos(9°)` `cos(9°) ≈ 0.9877` `r = 6370 0.9877 ≈ 6291.609 km` Apply small circle distance formula: `Distance = (α / 360°) * 2πr` `Distance = (8 / 360) 2 (22/7) * 6291.609` `Distance = (1 / 45) (44/7) 6291.609` `Distance = (1 / 45) * 197022.69 / 7` `Distance = 197022.69 / 315` `Distance ≈ 625.47 km` The distance flown by the aircraft is approximately 625.47 km.