If the height of a pole and the distance between the pole and a man standing nearby are equal, what would be the angle of elevation to the top of the pole?

Options

A. `60^(@)`
B. `90^(@)`
C. `30^(@)`
D. `45^(@)`

Answer

To solve the problem, we need to find the angle of elevation to the top of a pole, given that the height of the pole (h) and the distance from the pole to the man (d) are equal.

### Step-by-Step Solution:

1. **Define the Variables**:
Let the height of the pole be \( h \) and the distance from the man to the pole be \( d \). According to the problem, we have:
\[ h = d \]

2. **Draw a Right Triangle**:
When the man looks up at the top of the pole, he forms a right triangle with:
- The height of the pole as the opposite side (h).
- The distance from the man to the pole as the adjacent side (d).

3. **Use the Tangent Function**:
The angle of elevation \( x \) can be found using the tangent function, which is defined as:
\[ \tan(x) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{d} \]
Since \( h = d \), we can substitute:
\[ \tan(x) = \frac{h}{h} = 1 \]

4. **Find the Angle**:
We know that:
\[ \tan(45^\circ) = 1 \]
Therefore, if \( \tan(x) = 1 \), it follows that:
\[ x = 45^\circ \]

5. **Conclusion**:
The angle of elevation to the top of the pole is:
\[ \boxed{45^\circ} \]