How do we measure the distance between two points, A and B,...

How do we measure the distance between two points, A and B, on Earth? We measure along a circle with a center, C, at the center of Earth. The radius of the circle is equal to the distance from C to the surface. Use the fact that Earth is a sphere of radius equal to approximately 4000 miles to solve Exercises 93–96. If $\theta=10^{\circ},$ find the distance between $A$ and $B$ to the nearest mile.

How do we measure the distance between two points, A and B, on Earth? We measure along a circle with a center, C, at the center of Earth. The radius of the circle is equal to the distance from C to the surface. Use the fact that Earth is a sphere of radius equal to approximately 4000 miles to solve Exercises 93–96.
If $\theta=10^{\circ},$ find the distance between $A$ and $B$ to the nearest mile.

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Key Concepts

Arc Length Calculation

Arc length is determined by multiplying the radius of a circle by the central angle measured in radians. This approach is crucial for finding distances along curved paths, such as the surface of a sphere, and is based on the proportional relationship between the central angle and the circumference of the circle.

Conversion between Degrees and Radians

Since many mathematical formulas involving circles require angles in radians, converting from degrees to radians is essential. This conversion is done by multiplying the degree measure by ?/180, which allows for accurate use of formulas such as the one for calculating arc length.

Spherical Geometry

Spherical geometry deals with figures on the surface of a sphere. In this context, concepts such as measuring distances along great circles—circles whose centers coincide with that of the sphere—are central to accurately determining the shortest path between two points on the Earth's surface.

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