At a point 56 meters from the foot of a vertical aerial, the angle of elevation of the top of a flagpole is 52 degrees. What is the angle of elevation of its midpoint?

To solve the problem, we can break it down into steps and use trigonometry, particularly the tangent function, since we're dealing with angles of elevation and distances.

### Step 1: Understand the problem
- Let the height of the flagpole be \( h \).
- The distance from the point of observation to the foot of the flagpole is \( 56 \) meters.
- The angle of elevation to the top of the flagpole is \( 52^\circ \).

### Step 2: Find the height of the flagpole
Using the tangent function:
\[
\tan(52^\circ) = \frac{h}{56}
\]
To find the height \( h \), we rearrange the equation:
\[
h = 56 \times \tan(52^\circ)
\]

### Step 3: Calculate the height \( h \)
Using a calculator to find \( \tan(52^\circ) \):
\[
\tan(52^\circ) \approx 1.2799
\]
So,
\[
h \approx 56 \times 1.2799 = 71.6744 \text{ meters}
\]
The height of the flagpole is approximately \( 71.67 \) meters.

### Step 4: Find the height of the midpoint
The midpoint of the flagpole will be at \( \frac{h}{2} \):
\[
\text{Midpoint height} = \frac{71.67}{2} = 35.835 \text{ meters}
\]

### Step 5: Find the angle of elevation to the midpoint
Now, we use the tangent function again, but this time for the midpoint:
\[
\tan(\theta) = \frac{35.835}{56}
\]
\[
\tan(\theta) \approx 0.6409
\]
To find \( \theta \), we take the arctan (inverse tangent):
\[
\theta = \arctan(0.6409)
\]
Using a calculator:
\[
\theta \approx 32.66^\circ
\]

### Final Answer:
The angle of elevation to the midpoint of the flagpole is approximately \( 32.66^\circ \).

Would you like further details or have any questions?

### Related Questions:
1. How does changing the distance from the flagpole affect the angle of elevation?
2. What would be the angle of elevation to the top if the observer moved closer by 10 meters?
3. Can you find the height of the flagpole if the angle of elevation was 45 degrees?
4. How would the angle of elevation to the midpoint change if the flagpole's height was doubled?
5. How do you find the angle of elevation if only the distance and midpoint height are given?

**Tip:** When solving problems involving angles of elevation, remember to use the tangent function which relates the angle to the ratio of the opposite side (height) to the adjacent side (distance).

Learn how to calculate the angle of elevation to the midpoint of a flagpole when the angle to the top is known. This problem involves using trigonometric concepts such as the tangent function and properties of right triangles, suitable for Grades 9-11 students.

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