A radio antenna is attached to the top of a building. From a...

A radio antenna is attached to the top of a building. From a point 12.5 meters from the base of the building, on level ground, the angle of elevation of the bottom of the antenna is $47.2^{\circ}$ and the angle of elevation of the top is $51.8^{\circ}$. Find the height of the antenna.

Key Concepts

Angle of Elevation

The angle of elevation is the angle formed by the line of sight from an observer on level ground to a point above the horizontal. It is a fundamental concept in trigonometry, used to relate the height of an object and the distance from the observer via right triangle relationships.

Right Triangle Geometry

Right triangle geometry refers to the study of triangles where one angle is exactly 90 degrees. In many practical applications, such as determining heights and distances, a right triangle can be formed by the observer's line of sight, the horizontal distance, and the vertical height, making it a crucial tool for solving problems with angles of elevation.

Tangent Function

The tangent function in trigonometry is defined as the ratio of the side opposite an angle to the side adjacent in a right triangle. It is particularly useful in height and distance problems since it directly relates an angle of elevation to the unknown height when the horizontal distance is known.

Using Trigonometry to Calculate Height

Calculating height using trigonometry involves setting up a right triangle with the known horizontal distance and employing the tangent function with the given angle(s) of elevation. It often involves creating separate triangles for different measurements and finding the difference between calculated heights to derive the height of a specific segment, such as an antenna on top of a building.

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