To solve this problem, we need to determine the angle of elevation of the mid-point of the flagpole. We will use trigonometry, specifically the tangent function, since it relates the angle of elevation to the height and distance from the point of observation.

Given:

• The distance from the foot of the aerial to the point of observation: $275$ meters.
• The angle of elevation to the top of the flagpole: $16^\circ$.

Let's denote:

• $h$ as the height of the flagpole.
• $h/2$ as the height of the mid-point of the flagpole.

Step 1: Calculate the height of the flagpole $h$.

Using the tangent of the angle to find the height:

$\tan(16^\circ) = \frac{h}{275}$

$h = 275 \times \tan(16^\circ)$

Step 2: Calculate the angle of elevation to the mid-point.

Once we have the height $h$, the mid-point height will be $h/2$. The angle of elevation to the mid-point can be found using:

$\tan(\theta) = \frac{h/2}{275}$

We will solve these equations step by step. Let me calculate the values.The height of the flagpole is approximately $78.85$ meters. The angle of elevation to the mid-point of the flagpole is approximately $8.16^\circ$.

Would you like further details on the steps or have any questions?

Here are 5 related questions for you to explore:

1. How would the angle of elevation change if the distance from the flagpole increased?
2. What if the height of the flagpole were doubled? How would the mid-point angle change?
3. How do you find the angle of elevation if the point of observation is not horizontal?
4. What is the angle of elevation to the top if the flagpole height is reduced by 20%?
5. Can you calculate the horizontal distance needed for the angle of elevation to be exactly $45^\circ$?

Tip: When solving elevation problems, always sketch a right triangle to visualize relationships between angles and distances.