Radius of Earth A satellite is circling 112 miles above...

Radius of Earth A satellite is circling 112 miles above Earth, as shown in Figure $22 .$ When the satellite is directly above point $B$, angle $A$ is found to be $76.6^{\circ} .$ Use this information to find the radius of Earth.

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Key Concepts

Trigonometric Ratios in Right Triangles

This concept is about applying trigonometric functions, especially the cosine function, in right-angle triangle problems. When the relevant angle in the problem is known, the cosine ratio can relate the adjacent side (for example, the planet's radius) to the hypotenuse (the planet's radius plus the satellite's altitude). Using these relationships allows one to write and solve equations to find unknown lengths, such as the radius of the planet.

Circle Geometry and Tangency

This concept involves understanding the properties of circles, particularly how a tangent line to a circle is perpendicular to the radius at the point of tangency. In problems where a satellite is orbiting above a planet, the geometry of the situation often leads to a right triangle, where one vertex is at the center of the planet, one at the point directly below the satellite on the surface, and one at the satellite. Recognizing this relationship is crucial for setting up equations to solve for the planet's radius.

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