The height of a light house is 40 m. The angle of depression of a ship from the top of the light house is `60^(@)`. Find the distance of ship from the light house.

Answer

To solve the problem step by step, we will use trigonometric ratios and properties of angles.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a lighthouse (AB) that is 40 meters tall. The angle of depression from the top of the lighthouse to a ship (C) is 60 degrees. We need to find the horizontal distance (BC) from the base of the lighthouse (B) to the ship (C).

2. **Diagram Representation**:
Draw a vertical line representing the lighthouse (AB) with point A at the top and point B at the bottom. Draw a horizontal line from point B to point C, which represents the distance from the lighthouse to the ship. Draw a line from point A to point C. The angle of depression from A to C is 60 degrees.

3. **Identifying Angles**:
The angle of depression from A to C is 60 degrees. By the alternate interior angles theorem, the angle ACB (the angle at point C) is also 60 degrees.

4. **Using the Right Triangle**:
In triangle ABC, we have:
- AB = height of the lighthouse = 40 m (perpendicular)
- BC = distance from the lighthouse to the ship (base)
- AC = the line of sight from the top of the lighthouse to the ship (hypotenuse)

5. **Applying Trigonometric Ratios**:
We can use the tangent function, which is defined as the ratio of the opposite side (height of the lighthouse) to the adjacent side (distance from the lighthouse to the ship):
\[ \tan(60^\circ) = \frac{AB}{BC} \]
Here, \( AB = 40 \) m and \( BC = x \) (the distance we want to find).

6. **Substituting Values**:
We know that \( \tan(60^\circ) = \sqrt{3} \). Therefore, we can write:
\[ \sqrt{3} = \frac{40}{x} \]

7. **Cross Multiplying**:
To find \( x \), we cross-multiply:
\[ x \cdot \sqrt{3} = 40 \]

8. **Solving for x**:
Now, solve for \( x \):
\[ x = \frac{40}{\sqrt{3}} \]

9. **Rationalizing the Denominator**:
To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{3} \):
\[ x = \frac{40 \sqrt{3}}{3} \]

10. **Final Answer**:
The distance of the ship from the lighthouse is:
\[ x = \frac{40 \sqrt{3}}{3} \text{ meters} \]