HEIGHT Because of prevailing winds, a tree grew so that it...

HEIGHT Because of prevailing winds, a tree grew so that it was leaning $4^{\circ}$ from the vertical. At a point 40 meters from the tree, the angle of elevation to the top of the tree is $30^{\circ}$ (see figure). Find the height $h$ of the tree.

Key Concepts

Angle of Elevation

The angle of elevation is the angle between a horizontal line and the line of sight upward to an object. It is a key concept in indirect measurement problems because it, together with the horizontal distance, allows the calculation of an object’s height using trigonometric functions.

Trigonometric Ratios

Trigonometric ratios, such as sine, cosine, and tangent, define the relationships between the angles and sides of a right triangle. In problems involving an angle of elevation, these ratios are used to calculate unknown side lengths from known distances and angles, forming the backbone of most real-world trigonometry applications.

Right Triangle Trigonometry

This concept involves analyzing geometric problems by constructing right triangles, where one angle is 90°. It allows one to relate the sides of the triangle through trigonometric ratios and is fundamental when applying trigonometry to determine unknown lengths or heights from a given angle and distance.

Adjusting for Inclination

When an object is not perfectly vertical, its physical length differs from its vertical height. Adjusting for inclination involves using trigonometric functions, typically the cosine of the lean angle, to convert the measured length along the inclined object to its true vertical height. This adjustment is crucial in accurately solving problems with non-vertical structures.

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