Distance between Cities Find the distance in kilometers between each pair of cities, assuming they lie on the same north-south line. Assume the radius of Earth is 6400 km. See Example 2. Halifax, Nova Scotia , 45° N, and Buenos Aires, Argentina, 34° S

Verified step by step guidance

1. Identify the latitudes of the two cities: Halifax is at 45^\circ \text{N} and Buenos Aires is at 34^\circ \text{S}.
2. Since the cities are on the same north-south line, calculate the total angular difference in latitude: 45^\circ + 34^\circ.
3. Convert the total angular difference from degrees to radians using the conversion factor \( \pi \text{ radians} = 180^\circ \).
4. Use the formula for arc length on a circle: \( \text{Distance} = \text{radius} \times \text{angular difference in radians} \).
5. Substitute the radius of Earth (6400 km) and the angular difference in radians into the formula to find the distance.

Key Concepts

Great Circle Distance

The great circle distance is the shortest path between two points on the surface of a sphere. It is calculated using the spherical coordinates of the points, which in this case are the latitudes of Halifax and Buenos Aires. This concept is essential for determining the distance between cities that are aligned north-south, as it accounts for the curvature of the Earth.

Latitude

Latitude is a geographic coordinate that specifies the north-south position of a point on the Earth's surface. It is measured in degrees, with the equator at 0° and the poles at 90° N or S. Understanding latitude is crucial for calculating distances between cities located at different latitudes, as it directly influences the angle and distance traveled along the Earth's surface.

Earth's Radius

The Earth's radius is the average distance from the center of the Earth to its surface, approximately 6400 kilometers. This value is used in various calculations involving spherical geometry, including the great circle distance. Knowing the Earth's radius is vital for converting angular measurements (like degrees of latitude) into actual distances, which is necessary for solving the problem of finding the distance between Halifax and Buenos Aires.