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.
New York City, New York, 41° N, and Lima, Peru, 12° S

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Identify the latitudes of the two cities: New York City is at 41° N and Lima is at 12° S. Since they are on opposite sides of the equator, the total angular distance between them is the sum of their absolute latitudes.

Calculate the total central angle \( \theta \) between the two cities in degrees by adding the absolute values of their latitudes: \( \theta = 41^\circ + 12^\circ \).

Convert the central angle from degrees to radians because the arc length formula requires radians. Use the conversion formula: \( \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \).

Use the arc length formula to find the distance \( d \) between the two cities along the Earth's surface: \( d = r \times \theta_{radians} \), where \( r = 6400 \) km is the radius of the Earth.

Substitute the values into the formula and simplify to express the distance \( d \) in kilometers. This will give the distance between New York City and Lima along the north-south line.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Great Circle Distance on a Sphere

The shortest distance between two points on the surface of a sphere lies along the great circle connecting them. For points on the same meridian (north-south line), this distance is proportional to the difference in their latitudes multiplied by the Earth's radius.

Latitude and Angular Measurement

Latitude measures the angular distance north or south of the equator, expressed in degrees. To find the distance between two locations on the same meridian, calculate the absolute difference between their latitudes, converting degrees to radians if necessary for calculations.

Using Earth's Radius in Distance Calculations

The Earth's radius is essential for converting angular distances into linear distances. Multiplying the Earth's radius by the central angle (in radians) between two points on the surface gives the arc length, representing the actual distance between those points.

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