An aeroplane flies from point A (1015'S,370E) to a point B directly North of A. The arc AB subtends an angle of 45° at the centre of the earth. From B, the aeroplane flies due west to point C on longitude 230W. (Take the value of π as227 and radius of the earth as 6370km)
(a) (i) Find the latitude of
point B.
To find the latitude of point B, we need to use the fact that the arc AB subtends an angle of 45° at the center of the earth. We can use the formula for arc length:
s = rθ
where s is the length of the arc, r is the radius of the earth, and θ is the central angle in radians.
Since we know that the radius of the earth is 6370 km and the central angle is 45° (or π/4 radians), we can solve for the length of the arc AB:
s = 6370 km × (π/4) = 5001.7 km (rounded to one decimal place)
Next, we can use the coordinates of point A to find its distance from the equator. The latitude of point A is given by its distance from the equator, which is equal to the angle between the line connecting point A to the center of the earth and the equatorial plane. This angle can be found using the formula:
θ = arctan((rZ)/sqrt(X^2 + Y^2))
where X, Y, and Z are the coordinates of point A and r is the radius of the earth.
Plugging in the values for point A, we get:
θ = arctan((6370 km × -10.15)/sqrt((6370 km × 0.37)^2 + (6370 km × -10.15)^2)) = -27.7°
This tells us that point A is 27.7° south of the equator.
Since the arc AB lies directly north of point A, we know that the latitude of point B is equal to 90° minus the distance from the equator to point A. Therefore:
latitude of B = 90° - 27.7° = 62.3°
So the latitude of point B is 62.3 degrees.
(a) (ii) Find the distance of AB (correct to one decimal place)
To find the distance AB, we can use the law of cosines, which allows us to find the length of one side of a triangle given the lengths of the other two sides and the angle between them. In this case, we know the length of the first side (the radius of the earth), the angle between the first and second sides (45°), and the latitude of point B (62.3°), which allows us to find the length of the second side (AB).
Using the law of cosines, we get:
AB^2 = r^2 + s^2 - 2rs cos(θ)
where r is the radius of the earth, s is the length of the arc AB, and θ is the central angle between points A and B.
Plugging in the values we know, we get:
AB^2 = (6370 km)^2 + (5001.7 km)^2 - 2(6370 km)(5001.7 km) cos(45°) ≈ 63982944 km^2
Taking the square root of both sides, we get:
AB ≈ 7998.1 km
So the distance of AB is approximately 7998.1 kilometers.
(a) (iii) Find the longitude of point B.
Since point B lies directly north of point A, it has the same longitude as point A, which is 370° east. Therefore, the longitude of point B is 370° east.
(b) Find the distance of BC (correct to one decimal place)
To find the distance of BC, we need to find the length of the third side of the triangle formed by points B, C, and the center of the earth. We can use the law of cosines again, using the latitude of point B (62.3°) and the longitude of point C (230° west) to find the central angle between points B and C.
First, we need to convert the longitude of point C to an east longitude, which is 130° east. This is because 360° of longitude is equivalent to a complete circular rotation around the earth, and 230° west is the same as 130° east.
Next, we can use the formula for the central angle:
θ = arccos(sin(lat_B) sin(lat_C) + cos(lat_B) cos(lat_C) cos(long_B - long_C))
where lat_B and long_B are the latitude and longitude of point B, and lat_C and long_C are the latitude and longitude of point C.
Plugging in the values we know, we get:
θ = arccos(sin(62.3°) sin(0°) + cos(62.3°) cos(0°) cos(130° - 370°)) = 92.2°
Now we can use the law of cosines, using the radius of the earth, the length of AB, and the central angle between points B and C:
BC^2 = r^2 + AB^2 - 2rAB cos(θ)
Plugging in the values, we get:
BC^2 = (6370 km)^2 + (7998.1 km)^2 - 2(6370 km)(7998.1 km) cos(92.2°) ≈ 28329793 km^2
Taking the square root of both sides, we get:
BC ≈ 5321.6 km
So the distance of BC is approximately 5321.6 kilometers.