If the height of a vertical pole is equal to the length of its shadow on the ground, the angle of elevation of the sun is

Options

A. `0^(@)`
B. `30^(@)`
C. `45^(@)`
D. `60^(@)`

Explanation

To solve the problem, we need to find the angle of elevation of the sun when the height of a vertical pole is equal to the length of its shadow. Let's denote the height of the pole as \( h \) and the length of the shadow as \( h \) as well.

### Step-by-Step Solution:

1. **Understand the Setup**:
- Let \( AB \) be the vertical pole with height \( h \).
- Let \( BC \) be the shadow on the ground, which also has a length of \( h \).
- We need to find the angle of elevation \( \theta \) of the sun from point \( C \) (the tip of the shadow) to point \( A \) (the top of the pole).

2. **Identify the Right Triangle**:
- The triangle formed by points \( A \), \( B \), and \( C \) is a right triangle where:
- \( AB \) (height of the pole) is the opposite side to angle \( \theta \).
- \( BC \) (length of the shadow) is the adjacent side to angle \( \theta \).

3. **Use the Tangent Function**:
- The tangent of angle \( \theta \) is defined as the ratio of the opposite side to the adjacent side:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} = \frac{h}{h} \]

4. **Simplify the Expression**:
- Since both the height and the shadow length are equal:
\[ \tan(\theta) = \frac{h}{h} = 1 \]

5. **Find the Angle**:
- We know that \( \tan(45^\circ) = 1 \). Therefore, we can conclude that:
\[ \theta = 45^\circ \]

6. **Conclusion**:
- The angle of elevation of the sun is \( 45^\circ \).

### Final Answer:
The angle of elevation of the sun is \( 45^\circ \).

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