A telegraph post is bent at a point above the ground. Its top just touches the ground at a distance of 8√3 m from its foot and makes an angle of 30° with the horizontal. What is the height of the post in metres?

Introduction / Context:
This question is similar to the classic broken tree problem. A telegraph post is bent somewhere above the ground and its top touches the ground at some horizontal distance from the foot of the post. The broken part makes a known angle with the horizontal. We must determine the total original height of the post by applying basic trigonometry and Pythagoras theorem.

Given Data / Assumptions:

• The top of the post touches the ground at a distance of 8√3 m from its foot.
• The broken top makes an angle of 30 degrees with the horizontal.
• The post was originally vertical and straight.
• It is bent at a point h metres above the ground.
• The length of the broken part is L.

Concept / Approach:
When the post bends, the section above the bend becomes the hypotenuse of a right triangle whose base is the horizontal distance from the foot to the contact point and whose height is the vertical distance from the bend to the ground. Since the angle between the broken part and the ground is 30 degrees, we can use tan 30 = opposite / adjacent to find the height of the bend. Then, using Pythagoras theorem with the base 8√3 and this height, we compute the length of the broken part and add the two vertical segments to get the total height of the post.

Step-by-Step Solution:
Let h be the height of the bend above the ground. The horizontal distance from the foot to where the top touches the ground is 8√3 m. The angle between the broken part and the ground is 30 degrees. Using tan 30 = h / (8√3), we have 1 / √3 = h / (8√3). So h = (8√3) × (1 / √3) = 8 m. Let L be the length of the broken part (hypotenuse of the triangle). L² = h² + (8√3)² = 8² + 64 × 3 = 64 + 192 = 256. Hence L = 16 m. Total height of the telegraph post = h + L = 8 + 16 = 24 m.

Verification / Alternative check:
Quickly check with sine or cosine: For an angle of 30 degrees, cos 30 = adjacent / hypotenuse = (8√3) / 16 = (8√3) / 16 = √3 / 2, which matches cos 30. Also, sin 30 = opposite / hypotenuse = 8 / 16 = 1 / 2, which matches sin 30. Both checks confirm that our calculated lengths are consistent with the given angle.

Why Other Options Are Wrong:

• 12 m, 16 m, 18 m, 20 m: These values are all less than the sum of the two computed segments h and L and cannot satisfy both the angle condition and Pythagoras theorem simultaneously.
• Only 24 m correctly matches the geometry and trigonometry of the problem.

Common Pitfalls:
Sometimes students confuse the 30 degree angle as being at the foot of the post instead of at the point where the top touches the ground. Another mistake is to forget that the total height includes both the vertical segment up to the bend and the length of the broken segment. Drawing the right triangle and marking the angle helps clarify which side is which.

Final Answer:
The height of the telegraph post is 24 m.