A pole of height 6 m casts a shadow `2sqrt3` m long on the ground. Find the sun's elevation.

Answer

To solve the problem of finding the sun's elevation based on the height of the pole and the length of its shadow, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a pole of height 6 m and a shadow that is \(2\sqrt{3}\) m long. We need to find the angle of elevation of the sun, which we will denote as \(\theta\).

2. **Draw a Diagram**:
- Draw a vertical line representing the pole (AB) of height 6 m.
- Draw a horizontal line representing the shadow (BC) of length \(2\sqrt{3}\) m.
- The angle of elevation \(\theta\) is formed between the line of sight from the top of the pole (point A) to the end of the shadow (point C) and the horizontal line (BC).

3. **Identify the Triangle**:
- We can form a right triangle ABC where:
- AB = height of the pole = 6 m
- BC = length of the shadow = \(2\sqrt{3}\) m
- AC = the hypotenuse (the line of sight from the top of the pole to the end of the shadow).

4. **Use the Tangent Function**:
- In the right triangle ABC, we can use the tangent function to find the angle \(\theta\):
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} = \frac{6}{2\sqrt{3}} \]

5. **Simplify the Expression**:
- Simplifying \(\frac{6}{2\sqrt{3}}\):
\[ \tan(\theta) = \frac{6}{2\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \]

6. **Find the Angle \(\theta\)**:
- We know that \(\tan(60^\circ) = \sqrt{3}\), therefore:
\[ \theta = 60^\circ \]

7. **Conclusion**:
- The angle of elevation of the sun is \(60^\circ\).