Geometry

Question Analysis

The main focus of this problem is to use the horizontal distance from the helicopter's take-off point to the skyscraper and the height of the skyscraper to find the minimum angle of elevation the helicopter must have to avoid colliding with the skyscraper. We will apply trigonometric concepts in the right - triangle formed by the helicopter's path, the ground, and the height of the skyscraper.

Key Concept Explanation

1. Angle of Elevation: It is the angle between the horizontal line (where the helicopter takes off) and the line of the helicopter's path as it climbs. In a right - triangle, this angle relates the opposite side (height of the skyscraper) and the adjacent side (horizontal distance from the take-off point to the skyscraper) through trigonometric functions.

2. Trigonometric Functions in Right - Triangles: For an acute angle  in a right - triangle, . This formula will be used to calculate the angle of elevation.

Step - by - Step Solution

1. Let the height of the skyscraper be  feet and the horizontal distance from the helicopter's take-off point to the skyscraper be  feet.

We know that , where  is the angle of elevation.

Substitute  and  into the formula: .

To find , we take the inverse tangent (arctan) of . So,

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