**At a point 56 m from the foot of a vertical aerial, the angle of elevation of the top of a flagpole is 24o. What is the angle of elevation of its mid-point?**

To solve this problem, we can break it into two parts:

1. **Determine the height of the flagpole**: 
   The given information tells us the horizontal distance from the foot of the flagpole (56 m) and the angle of elevation to the top (24°). Using the trigonometric tangent function (since tangent relates an angle to the ratio of opposite side over adjacent side in a right triangle), we can find the height of the flagpole.

   \[
   \tan(24^\circ) = \frac{h_{\text{top}}}{56}
   \]

   Rearranging this equation to solve for \( h_{\text{top}} \) (the height of the flagpole):

   \[
   h_{\text{top}} = 56 \times \tan(24^\circ)
   \]

2. **Find the angle of elevation of the mid-point**: 
   The height of the mid-point of the flagpole is half of the total height:

   \[
   h_{\text{mid}} = \frac{h_{\text{top}}}{2}
   \]

   To find the angle of elevation to the mid-point, we again use the tangent function:

   \[
   \tan(\theta_{\text{mid}}) = \frac{h_{\text{mid}}}{56}
   \]

   Solving for \( \theta_{\text{mid}} \):

   \[
   \theta_{\text{mid}} = \tan^{-1} \left( \frac{h_{\text{mid}}}{56} \right)
   \]

### Step-by-step solution:

1. Calculate the height of the flagpole:
   \[
   h_{\text{top}} = 56 \times \tan(24^\circ) = 56 \times 0.4452 = 24.9312 \, \text{m}
   \]

2. The height of the mid-point:
   \[
   h_{\text{mid}} = \frac{24.9312}{2} = 12.4656 \, \text{m}
   \]

3. Find the angle of elevation to the mid-point:
   \[
   \tan(\theta_{\text{mid}}) = \frac{12.4656}{56} = 0.2226
   \]

   \[
   \theta_{\text{mid}} = \tan^{-1}(0.2226) = 12.6^\circ
   \]

Thus, the angle of elevation to the mid-point of the flagpole is approximately **12.6°**.

Would you like any further details or have any questions?

### Related Questions:
1. What would the angle of elevation be if the flagpole were 10 meters taller?
2. How would the problem change if the horizontal distance were 80 meters instead of 56 meters?
3. How is the tangent function typically used in solving elevation problems?
4. What would be the angle of elevation to the three-quarters height of the flagpole?
5. How would this calculation change if the pole were on a sloped hill?

### Tip:
In elevation problems, the tangent of the angle is often your best friend since it relates heights and distances directly.

At a point 56 m from the foot of a vertical aerial, the angle of elevation of the top of a flagpole is 24°. What is the angle of elevation of its mid-point?

Learn how to calculate the angle of elevation of the mid-point of a flagpole using trigonometry. In this problem, the horizontal distance from the foot of the flagpole is 56 meters, and the angle of elevation to the top is 24°. This solution involves key trigonometric concepts and formulas suitable for students in Grades 9-10.

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