An air craft flies from a point A (1°15'S,37°E) to a point B directly North of A . The arc subtends an angle of 48° at the centre of the earth. From B the aeroplane flies due west to a point C on the longitude 23°W. Take radius of the earth as 6370km:
(a) . State the location of B
(b). Find the distance in Km traveled by the aeroplane between B and C
(a) To determine the location of B, we need to draw a diagram:
A
/ \
/ \
/ \
/ \
/ \
O----B-----C
Here, A is the starting point (1°15'S, 37°E) and C is the destination (23°W). The arc subtending an angle of 48° at the center of the earth is the arc AB. We need to find the location of B, which is directly north of A.
Since the angle subtended at the center of the earth is 48°, the angle subtended at the circumference of the earth (at point A) is also 48°. Therefore, the angle between the equator and the line AB is (90° - 48°) = 42°.
We can use this angle to find the latitude of point B. Using the sine rule, we have:
sin 42° / 6370 = sin θ / 6370
where θ is the angle between the equator and the line AB at point A. Solving for θ, we get:
θ = sin⁻¹(sin 42° / 6370 * 6370) = 20.561°
Since point B is directly north of A, its longitude is the same as A's longitude, i.e. 37°E. Therefore, the location of B is (20.561°N, 37°E).
(b) To find the distance traveled by the airplane between B and C, we can use the cosine rule. Let D be the angle between the lines BC and OB.
Using the cosine rule, we have:
cos D = cos 90° * cos 23°W + sin 90° * sin 23°W * cos 48°
= sin 23°W * cos 48°
cos D = (sin 23° * cos 48°) = 0.3657
D = cos⁻¹(0.3657) = 68.428°
Since the angle subtended at the center of the earth is 68.428°, the angle subtended at the circumference of the earth (at point B) is also 68.428°.
Using the sine rule, we can find the distance traveled by the airplane between B and C:
sin 68.428° / 6370 = sin θ / x
where x is the distance between B and C. Solving for x, we get:
x = sin⁻¹(sin 68.428° / 6370 * x) = 6818 km
Therefore, the distance traveled by the airplane between B and C is approximately 6818 km.
To determine the location of point B, we need to calculate its latitude and longitude.
Given:
Point A: Latitude = 1°15'S, Longitude = 37°E
Arc subtends an angle of 48° at the center of the Earth
(a) Finding the latitude of point B:
Since the arc subtends an angle of 48° at the center of the Earth, the latitude of B is equal to the angle.
Latitude of B = 48°
(b) Finding the longitude of point B:
Given that point B is directly north of point A, the longitude of B will be the same as point A.
Longitude of B = 37°E
Therefore, the location of point B is 48°N, 37°E.
Now, let's calculate the distance traveled by the airplane between B and C.
Given:
Radius of the Earth = 6370 km
Longitude of C = 23°W
To find the distance between two points on the same latitude, we can use the formula:
Distance = Circumference of the Earth * (Difference in longitudes / 360°)
Circumference of the Earth = 2π * Radius of the Earth
Circumference = 2 * 3.141 * 6370 km = 40075.16 km (approximately)
To find the difference in longitudes, we need to calculate the absolute value of the longitude difference between C and B:
Difference in longitudes = |Longitude of B - Longitude of C|
Difference in longitudes = |37°E - 23°W|
Converting 23°W to degrees east: 180° - 23° = 157°E
Difference in longitudes = |37°E - 157°E|
Difference in longitudes = 120°
Now, using the formula mentioned above, we can calculate the distance traveled between B and C:
Distance = 40075.16 km * (120° / 360°)
Distance = 40075.16 km * 1/3
Distance = 13358.39 km (approximately)
Therefore, the distance traveled by the airplane between B and C is approximately 13358.39 km.