A player sitting on the top of a tower of height 40 m observes the angle of depression of a ball lying on the ground is `60^@`. The distance between the foot of the tower and the ball is

Options

A. `50/sqrt3m`
B. `40/sqrt3m`
C. `40sqrt3m`
D. `50sqrt3m`

Answer

To solve the problem step by step, we can follow these instructions:

1. **Understand the Problem**: We have a tower of height 40 meters, and the angle of depression from the top of the tower to a ball on the ground is 60 degrees. We need to find the horizontal distance from the foot of the tower to the ball.

2. **Draw a Diagram**:
- Let the tower be represented as a vertical line segment AB, where A is the top of the tower and B is the foot of the tower.
- The height of the tower AB = 40 m.
- Let C be the position of the ball on the ground.
- The angle of depression from A to C is 60 degrees.

3. **Identify Angles**:
- The angle of depression is the angle formed by the horizontal line from A and the line AC (from A to C).
- Therefore, the angle CAB (the angle at B) is complementary to the angle of depression. Thus, angle CAB = 90° - 60° = 30°.

4. **Use Trigonometric Ratios**:
- In triangle ABC, we can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side.
- Here, the opposite side is the height of the tower (AB = 40 m), and the adjacent side is the distance from the foot of the tower to the ball (BC, which we will denote as x).

5. **Set Up the Equation**:
\[ \tan(30^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} = \frac{40}{x} \]

6. **Substitute the Value of Tan(30°)**:
- We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\).
\[ \frac{1}{\sqrt{3}} = \frac{40}{x} \]

7. **Cross-Multiply to Solve for x**:
\[ x = 40 \cdot \sqrt{3} \]

8. **Final Calculation**:
- To express the answer in a simplified form, we can rationalize it:
\[ x = \frac{40 \sqrt{3}}{3} \text{ meters} \]

Thus, the distance between the foot of the tower and the ball is \( \frac{40 \sqrt{3}}{3} \) meters.