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 find 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. Let's denote the height of the pole as \(h\).

### Step-by-Step Solution:

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

2. **Understand the Right Triangle**:
- When the sun casts a shadow, we can visualize a right triangle formed by:
- The height of the pole (perpendicular) = \(h\)
- The length of the shadow (base) = \(\sqrt{3}h\)
- The angle of elevation of the sun = \(\theta\)

3. **Use the Tangent Function**:
- In a right triangle, the tangent of an angle is defined as the ratio of the opposite side to the adjacent side. Therefore, we can write:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{\sqrt{3}h} \]

4. **Simplify the Equation**:
- The \(h\) in the numerator and denominator cancels out:
\[ \tan(\theta) = \frac{1}{\sqrt{3}} \]

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

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

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