A surveyor measures the angle of elevation of the top of a...

A surveyor measures the angle of elevation of the top of a perpendicular building as $19^{\circ}$. He moves $120 \mathrm{~m}$ nearer the building and finds the angle of elevation is now $47^{\circ}$. Determine the height of the building.

Key Concepts

Angle of Elevation

This is the angle formed between the horizontal line of sight and the line of sight directed upwards towards an object. It is a fundamental concept in trigonometry, especially in problems involving heights and distances, where measurements from ground level are used to determine the heights of objects like buildings or mountains.

Right Triangle Trigonometry

Right triangle trigonometry involves the study of the relationships between the angles and sides of right-angled triangles. The concepts and formulas in this area are essential for solving problems where one needs to relate horizontal distances and vertical heights, which is a common scenario in surveying and architecture.

Tangent Function

The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. This function is particularly useful in problems involving angles of elevation because it directly relates the height of an object (opposite side) to the horizontal distance from the observer (adjacent side).

System of Equations

When multiple measurements or observations provide different relationships between the same set of unknowns, a system of equations can be set up to solve for these unknown quantities. In the context of height determination using angles of elevation from two separate positions, the equations derived from the tangent function can be solved simultaneously to determine the height of the object and its distance from the observation points.

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