Geometry

Question Analysis

The main focus of this problem is to use the horizontal distance from the player to the fence and the height of the fence to find the minimum angle of elevation at which the ball must be thrown to clear the fence. We will apply trigonometric concepts in the right - triangle formed by the ball's path, the ground, and the height of the fence.

Key Concept Explanation

1. Angle of Elevation: It is the angle between the horizontal line (where the player throws the ball) and the path of the ball as it travels towards the fence. In a right - triangle, this angle relates the opposite side (height of the fence) and the adjacent side (horizontal distance from the player to the fence) 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 fence be feet and the horizontal distance from the player to the fence 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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