An observer who is 1.62m tall is 45m away from a pole . The angle of elevation of the top of the pole from his eyes is `30^(@)`. The height ( in m ) of the pole is closest to `:`

Options

A. 26.2
B. 27.6
C. 25.8
D. 26.8

Answer

To solve the problem, we will follow these steps:

1. **Understand the Problem**: We have an observer who is 1.62 meters tall, standing 45 meters away from a pole. The angle of elevation from the observer's eyes to the top of the pole is 30 degrees. We need to find the height of the pole.

2. **Draw a Diagram**:
- Let point O be the observer's position, point M be the observer's eyes, and point A be the top of the pole.
- The height of the observer's eyes (M) is 1.62 meters above the ground.
- The distance from the observer to the pole (OM) is 45 meters.
- The angle of elevation (∠AMO) is 30 degrees.

3. **Identify the Triangles**:
- We will focus on triangle AMO, where:
- AM is the height from the observer's eyes to the top of the pole.
- OM is the horizontal distance from the observer to the pole (45 meters).

4. **Use Trigonometry**:
- In triangle AMO, we can use the tangent function since we have the opposite side (AM) and the adjacent side (OM).
- The formula for tangent is:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
- Here, \(\theta = 30^\circ\), so:
\[ \tan(30^\circ) = \frac{AM}{OM} \]
- We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\).

5. **Set Up the Equation**:
- Plugging in the values:
\[ \frac{1}{\sqrt{3}} = \frac{AM}{45} \]
- Rearranging gives:
\[ AM = 45 \cdot \frac{1}{\sqrt{3}} = \frac{45}{\sqrt{3}} \approx 25.98 \text{ meters} \]

6. **Calculate the Total Height of the Pole**:
- The total height of the pole (AB) is the sum of the height from the observer's eyes to the top of the pole (AM) and the height of the observer (MB):
\[ AB = AM + MB = 25.98 + 1.62 = 27.60 \text{ meters} \]

7. **Final Answer**:
- The height of the pole is approximately 27.6 meters.

### Summary of Steps:
1. Understand the problem and identify key points.
2. Draw a diagram to visualize the situation.
3. Identify the relevant triangle and apply trigonometric functions.
4. Set up the equation using the tangent function.
5. Solve for the height from the observer's eyes to the top of the pole.
6. Add the observer's height to find the total height of the pole.
7. State the final answer.