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To solve this problem, let’s break it down into steps.

### Problem Analysis

Given:
- Distance from the foot of the flagpole (or aerial) to the observation point: \( 275 \) m.
- Angle of elevation to the top of the flagpole: \( 16^\circ \).
- We are asked to find the angle of elevation to the midpoint of the flagpole.

### Solution

1. **Let the height of the flagpole be \( h \).**
   
   Since we know the distance to the flagpole and the angle to its top, we can find \( h \) using trigonometry. The tangent of the angle of elevation is given by:
   \[
   \tan(16^\circ) = \frac{h}{275}
   \]
   
   Solving for \( h \):
   \[
   h = 275 \times \tan(16^\circ)
   \]

2. **Calculate the Height \( h \):**
   
   Using \( \tan(16^\circ) \approx 0.2867 \),
   \[
   h \approx 275 \times 0.2867 = 78.8425 \text{ m}
   \]

3. **Determine the Height of the Midpoint:**
   
   The midpoint of the flagpole is at \( \frac{h}{2} \):
   \[
   \frac{h}{2} \approx \frac{78.8425}{2} = 39.42125 \text{ m}
   \]

4. **Find the Angle of Elevation to the Midpoint:**
   
   Now, we want to find the angle \( \theta \) where the height is \( 39.42125 \) m and the horizontal distance remains \( 275 \) m.

   Using the tangent function:
   \[
   \tan(\theta) = \frac{39.42125}{275}
   \]
   
   \[
   \theta = \tan^{-1} \left( \frac{39.42125}{275} \right)
   \]

5. **Calculate \( \theta \):**
   
   \[
   \frac{39.42125}{275} \approx 0.1434
   \]
   
   \[
   \theta \approx \tan^{-1}(0.1434) \approx 8.2^\circ
   \]

### Answer
The angle of elevation to the midpoint of the flagpole is approximately \( 8.2^\circ \).

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### Related Questions

1. How do you solve similar problems when the object is tilted instead of vertical?
2. What would happen to the angle if the flagpole’s height were doubled?
3. How does the distance from the observation point affect the angle of elevation?
4. Could we solve this problem if only the height of the midpoint was given?
5. How is trigonometry used to calculate distances in real-life applications?

### Tip
Always label points in a trigonometric problem to visualize relationships between angles and distances clearly!

At a point 275 m from the foot of a vertical aerial, the angle of elevation of the top of a flagpole is 16°. What is the angle of elevation of its mid-point?

Learn how to calculate the angle of elevation to the midpoint of a flagpole given a point 275 m from its base with a 16° angle of elevation to its top. This problem uses trigonometry and the tangent function, ideal for Grades 9-10 students.

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