The angle of elevation of the sun when the length of the shadow of a pole is `sqrt(3)` times the height of the pole is

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `15^(@)`

Answer

To solve the problem, we need to determine the angle of elevation of the sun given that the length of the shadow of a pole is \(\sqrt{3}\) times the height of the pole.

### Step-by-Step Solution:

1. **Define Variables:**
Let the height of the pole be \( h \). According to the problem, the length of the shadow is given as \( \sqrt{3}h \).

2. **Set Up the Right Triangle:**
In the right triangle formed by the pole, the shadow, and the line from the top of the pole to the tip of the shadow, we have:
- The height of the pole (opposite side) = \( h \)
- The length of the shadow (adjacent side) = \( \sqrt{3}h \)

3. **Use the Tangent Function:**
The tangent of the angle of elevation \( \theta \) 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:**
The \( h \) in the numerator and denominator cancels out:
\[ \tan(\theta) = \frac{1}{\sqrt{3}} \]

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. **Final Answer:**
The angle of elevation of the sun is \( 30^\circ \).