The angle of depression of a ship from the top a tower of height 50 m is `30^(@)`. Find the horizontal distance between the ship and tower.

Answer

To solve the problem, we will follow these steps:

1. **Understand the Problem**: We have a tower of height 50 meters, and we need to find the horizontal distance from the base of the tower to a ship, given that the angle of depression from the top of the tower to the ship is \(30^\circ\).

2. **Draw a Diagram**:
- Draw a vertical line representing the tower (let's call it AC, where A is the top of the tower and C is the base).
- Mark point B as the position of the ship.
- The angle of depression from point A to point B is \(30^\circ\).
- The horizontal distance from the base of the tower (C) to the ship (B) will be represented as BC.

3. **Identify the Triangle**:
- In triangle ABC, angle ACB is \(30^\circ\) (alternate interior angle).
- AC (the height of the tower) is 50 meters (the perpendicular side).
- BC is the horizontal distance we need to find (the base).

4. **Use Trigonometric Ratios**:
- We can use the tangent function, which relates the angle of a right triangle to the opposite side and the adjacent side.
- The formula is:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]
- Here, \(\theta = 30^\circ\), the opposite side is AC (50 m), and the adjacent side is BC.

5. **Set Up the Equation**:
- Substitute the known values into the tangent formula:
\[ \tan(30^\circ) = \frac{AC}{BC} = \frac{50}{BC} \]

6. **Calculate \(\tan(30^\circ)\)**:
- We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\).

7. **Substitute and Solve for BC**:
- Now we can set up the equation:
\[ \frac{1}{\sqrt{3}} = \frac{50}{BC} \]
- Cross-multiplying gives:
\[ BC = 50 \sqrt{3} \]

8. **Final Answer**:
- The horizontal distance between the ship and the tower is \(50\sqrt{3}\) meters.

### Summary of Steps:
1. Understand the problem and draw a diagram.
2. Identify the triangle and the sides involved.
3. Use the tangent function to relate the sides.
4. Substitute known values and solve for the unknown distance.