Assuming that the Earth is spherical and recalling that latitudes range from 0° at the Equator to 90° N at the North Pole, how far apart, measured on the Earth's surface, are Copenhagen (55.67° N latitude), and Rome (41.90° N latitude)? The two cities lie on approximately the same longitude. Do not neglect the curvature of the Earth in determining this distance.
The answer is supposed to be 1.53e+06 m. How do I find this answer?
Convert the latitude difference of the two cities from degrees to radians. Then multiply that angle by the radius of the Earth.
Arc length = radius * (angle in radians)
To find the distance between Copenhagen and Rome, we can use the formula for calculating the distance between two points on the surface of a sphere:
distance = radius of the Earth × central angle
1. First, convert the latitude values to radians. Since the formula requires angles in radians, we can convert the latitude values from degrees to radians.
Latitude of Copenhagen = 55.67° N = 55.67 × (π/180) radians.
Latitude of Rome = 41.90° N = 41.90 × (π/180) radians.
2. Since the cities have approximately the same longitude, the central angle between them is simply the difference in latitudes.
Central angle = |Latitude of Copenhagen - Latitude of Rome|.
3. The radius of the Earth is approximately 6,371 kilometers or 6,371,000 meters.
4. Plug in the values into the formula:
distance = (radius of the Earth) × (central angle)
= 6,371,000 × |(55.67 × (π/180)) - (41.90 × (π/180))|
5. Calculate the absolute difference between the latitude values and multiply it by the radius of the Earth:
distance = 6,371,000 × |(55.67 × (π/180)) - (41.90 × (π/180))|
6. Simplify the equation:
distance = 6,371,000 × |(55.67 - 41.90) × (π/180)|
7. Calculate the final value:
distance = 6,371,000 × |13.77 × (π/180)|
8. Use the approximation of π as 3.14159:
distance = 6,371,000 × |13.77 × (3.14159/180)|
distance = 1.53e+06 meters (approximately)
Therefore, the distance between Copenhagen and Rome, measured on the Earth's surface, is approximately 1.53e+06 meters.
To find the distance between Copenhagen (55.67° N latitude) and Rome (41.90° N latitude) while accounting for the curvature of the Earth, you can use the Haversine formula. This formula calculates the distance between two points on a sphere (in this case, the Earth) using their latitudes and longitudes.
Here's how you can use the Haversine formula to find the distance between Copenhagen and Rome:
1. Convert the latitude values from degrees to radians. Since the Haversine formula uses radians as input, we need to convert the latitudes of Copenhagen (55.67° N) and Rome (41.90° N) to radians.
- Convert 55.67° to radians:
- Radians = 55.67 × π / 180
- Convert 41.90° to radians:
- Radians = 41.90 × π / 180
2. Calculate the central angle between the two latitudes. The central angle between the two latitudes can be calculated using the formula:
- Central angle = |latitude1 - latitude2|
3. Calculate the arc length based on the central angle. The arc length can be calculated using the formula:
- Arc length = radius of the Earth × central angle
Note: The radius of the Earth is approximately 6,371 kilometers (or 6,371,000 meters).
4. Convert the arc length to meters (if necessary). Depending on the unit used for the radius of the Earth, you may need to convert the arc length to meters.
- 1 kilometer = 1000 meters
5. Round the final result to the appropriate number of decimal places. The answer provided is 1.53e+06 meters, which is written in scientific notation. This is equivalent to 1,530,000 meters or 1.53 million meters.
Following these steps should allow you to calculate the distance between Copenhagen and Rome on the Earth's surface, taking into account the curvature of the Earth.