Solve each problem.See Examples 3 and 4. Distance from the Ground to the Top of a Building The angle of depression from the top of a building to a point on the ground is 32°30'. How far is the point on the ground from the top of the building if the building is 252 m high?

Verified step by step guidance

1. Identify the right triangle formed by the building height, the distance from the building's base to the point on the ground, and the line of sight from the top of the building to the point on the ground.
2. Recognize that the angle of depression from the top of the building to the point on the ground is equal to the angle of elevation from the point on the ground to the top of the building, which is 32°30'.
3. Label the height of the building as the opposite side of the angle (252 m), and the distance from the point on the ground to the base of the building as the adjacent side of the angle.
4. Use the tangent trigonometric ratio, which relates the opposite side and adjacent side in a right triangle: \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).
5. Set up the equation \(\tan(32^{\circ}30') = \frac{252}{x}\), where \(x\) is the distance from the point on the ground to the base of the building, then solve for \(x\) by rearranging the equation to \(x = \frac{252}{\tan(32^{\circ}30')}\).

Key Concepts

Angle of Depression

The angle of depression is the angle formed between the horizontal line from the observer's eye and the line of sight down to an object. In this problem, it helps relate the height of the building to the horizontal distance on the ground.

Right Triangle Trigonometry

The scenario forms a right triangle where the building height is one leg, the horizontal distance is the other leg, and the line of sight is the hypotenuse. Trigonometric ratios like tangent relate angles to side lengths in such triangles.

Using Tangent Function

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Here, tan(32°30') = height / horizontal distance, allowing calculation of the unknown distance from the building to the point on the ground.