At an instant, the length of the shadow of a pole is `sqrt3` times the height of the pole. The angle of elevation of the Sun at that moment is

Options

A. `75^@`
B. `30^@`
C. `45^@`
D. `60^@`

Answer

To solve the problem, we will use trigonometric relationships in a right triangle formed by the pole and its shadow.

### Step-by-Step Solution:

1. **Define the Variables**:
Let the height of the pole be \( h \). According to the problem, the length of the shadow of the pole is \( \sqrt{3} \) times the height of the pole. Therefore, the length of the shadow can be expressed as:
\[ \text{Length of shadow} = \sqrt{3} \cdot h \]

2. **Draw the Right Triangle**:
We can visualize a right triangle where:
- The height of the pole (AB) is the opposite side (perpendicular).
- The length of the shadow (BC) is the adjacent side (base).
- The angle of elevation of the sun (θ) is the angle between the ground and the line of sight to the top of the pole.

3. **Set Up the Tangent Function**:
The tangent of the angle of elevation (θ) is given by the ratio of the opposite side to the adjacent side:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{\sqrt{3}h} \]

4. **Simplify the Expression**:
We can simplify the expression for tangent:
\[ \tan(\theta) = \frac{h}{\sqrt{3}h} = \frac{1}{\sqrt{3}} \]

5. **Find the Angle**:
We know from trigonometric values that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
Therefore, we can conclude that:
\[ \theta = 30^\circ \]

6. **Conclusion**:
The angle of elevation of the sun at that moment is:
\[ \boxed{30^\circ} \]