The length of the shadow of a pole at a time is `sqrt3`times its height.Find the sun's altitude

Answer

To solve the problem, we need to find the sun's altitude given that the length of the shadow of a pole is \( \sqrt{3} \) times its height.

### 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 \( L \) is given by:
\[ L = \sqrt{3} \times H \]

2. **Understanding 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 can identify:
- The height of the pole as the opposite side (perpendicular).
- The length of the shadow as the adjacent side (base).
- The angle of elevation of the sun as \( \theta \).

3. **Using the Tangent Function**:
The tangent of the angle \( \theta \) is defined as the ratio of the opposite side to the adjacent side:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{H}{L} \]
Substituting the expression for \( L \):
\[ \tan(\theta) = \frac{H}{\sqrt{3}H} \]

4. **Simplifying the Expression**:
Since \( H \) is in both the numerator and denominator, we can cancel it out:
\[ \tan(\theta) = \frac{1}{\sqrt{3}} \]

5. **Finding the Angle \( \theta \)**:
To find \( \theta \), we need to determine the angle whose tangent is \( \frac{1}{\sqrt{3}} \). From trigonometric ratios, we know:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
Therefore, we have:
\[ \theta = 30^\circ \]

6. **Conclusion**:
The sun's altitude is \( 30^\circ \).