If the length of shadow of a pole on a level ground is twice the length of that pole, the angle of elevation of the Sun is

Options

A. `60^(@)`
B. `30^(@)`
C. `45^(@)`
D. None of these

Answer

To solve the problem, we need to find the angle of elevation of the Sun given that the length of the shadow of a pole is twice the length of the pole itself.

### Step-by-Step Solution:

1. **Define the Variables:**
Let the height of the pole be \( d \). According to the problem, the length of the shadow is twice the height of the pole, so the length of the shadow is \( 2d \).

2. **Draw a Diagram:**
Visualize the situation: you have a vertical pole of height \( d \) and a horizontal shadow of length \( 2d \). The angle of elevation of the Sun is the angle \( \theta \) formed between the line from the top of the pole to the tip of the shadow and the horizontal ground.

3. **Identify the Right Triangle:**
This situation forms a right triangle where:
- The height of the pole \( d \) is the opposite side.
- The length of the shadow \( 2d \) is the adjacent side.

4. **Use the Tangent Function:**
The tangent of the angle \( \theta \) can be expressed as:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{d}{2d} \]

5. **Simplify the Expression:**
Simplifying the fraction gives:
\[ \tan(\theta) = \frac{1}{2} \]

6. **Find the Angle \( \theta \):**
To find \( \theta \), we take the arctangent (inverse tangent) of \( \frac{1}{2} \):
\[ \theta = \tan^{-1}\left(\frac{1}{2}\right) \]

7. **Conclusion:**
The angle of elevation of the Sun is \( \tan^{-1}\left(\frac{1}{2}\right) \).