Broken pole touches ground at 30°: A vertical pole breaks at some point, and the top touches the ground at a point 21 m from the foot of the pole. The broken top makes an angle of 30° with the ground. Find the original total height of the pole.

Introduction / Context:
After breaking, the upper segment forms the hypotenuse of a right triangle with the ground. The remaining standing segment equals the vertical component of that hypotenuse. The original height is the sum of the two segments.

Given Data / Assumptions:

• Horizontal distance from foot to tip = 21 m.
• Inclination of top with ground = 30°.
• Let broken top length be L; remaining vertical be y.

Concept / Approach:
Resolve L into horizontal and vertical components: L cos 30° = 21 and L sin 30° = y. Total original height H = y + L (standing part + broken top length).

Step-by-Step Solution:

Verification / Alternative check:
Numeric: √3 ≈ 1.732 → H ≈ 36.37 m; geometry checks out with the given angle and reach of 21 m.

Why Other Options Are Wrong:
21 m is just the horizontal reach; 21/√3 is only the standing part; “None” is unnecessary since a clean expression exists.

Common Pitfalls:
Adding vertical components only; forgetting that the original top length equals the current slanted length; mixing degrees/radians.

Final Answer:
21 √3 m