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

Introduction / Context:
This is another broken-pole or broken-post type of height and distance question. The telegraph post was originally vertical, but after breaking at some point, its top touches the ground forming a right triangle. Using the given angle and the distance from its base, we can reconstruct both the remaining vertical part and the broken section to recover the original height.

Given Data / Assumptions:

• The telegraph post is originally vertical on level ground.
• After breaking at some height, its top touches the ground at a horizontal distance of 8√3 m from the base.
• The broken part makes an angle of 30° with the ground.
• The broken part remains straight and attached to the stump.
• We must find the original total height of the post.

Concept / Approach:
The broken top is the hypotenuse of a right triangle with base 8√3 m and angle at the ground of 30°. From this, we find the length of the broken part using cos(30°). The vertical leg of this triangle is the height at which the post broke. The original height is then the sum of this vertical leg and the broken part, since that top piece was previously vertical as well.

Step-by-Step Solution:
Let L be the length of the broken part.Base (horizontal distance) = 8√3 m.Angle between broken part and ground = 30°.cos(30°) = base / hypotenuse = (8√3) / L.cos(30°) = √3 / 2, so √3 / 2 = (8√3) / L.Cancel √3 to get: 1 / 2 = 8 / L ⇒ L = 16 m.Vertical height of the breaking point = L * sin(30°).sin(30°) = 1 / 2 ⇒ height of break = 16 * 1 / 2 = 8 m.Original height of the post = vertical stump + broken part = 8 + 16 = 24 m.

Verification / Alternative check:
Check the right triangle with sides: hypotenuse 16 m, angle 30° at base, base 8√3 m, vertical leg 8 m. Using sin(30°) = opposite / hypotenuse gives 8 / 16 = 1 / 2, and cos(30°) = adjacent / hypotenuse gives (8√3) / 16 = √3 / 2, so the geometry is consistent and confirms the result.

Why Other Options Are Wrong:
12 m, 16 m, 18 m, 20 m: Each is either the stump height, the hypotenuse length, or an arbitrary value. None equal stump + broken part, which must be 8 + 16.

Common Pitfalls:
One common error is to take only the vertical leg (8 m) or only the broken length (16 m) as the height of the post. Another is confusing which side is opposite and which is adjacent to the 30° angle, leading to using sine where cosine is required, or vice versa. Always sketch the triangle to avoid such mistakes.

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