Two men stand on the same side of a vertical pillar that is 75 m high and observe the angle of elevation of the top of the pillar to be 30° and 60°, respectively. What is the distance (in metres) between the two men?

Introduction:
This is a standard height and distance problem using trigonometry. Two observers stand on the same side of a vertical pillar and see its top under different angles of elevation. Using tangent relations for right-angled triangles, we can find each man’s horizontal distance from the pillar and then determine the distance between them.

Given Data / Assumptions:

• Height of the pillar = 75 m.
• First man's angle of elevation = 60°.
• Second man's angle of elevation = 30°.
• Both are on the same straight line on one side of the pillar.
• The ground is level and the pillar is perfectly vertical.

Concept / Approach:
For a vertical object of height H and a point on level ground at distance d from its foot, the angle of elevation θ satisfies:
tan(θ) = H / d.Using this, we can find the distances of each man from the pillar. The man with the larger angle (60°) is closer; the man with the smaller angle (30°) is farther. The difference of these distances gives the distance between them.

Step-by-Step Solution:
Step 1: Let d₁ be the distance of the man seeing 60°, and d₂ be the distance of the man seeing 30°.Step 2: Using tan(60°) = √3: √3 = 75 / d₁ ⇒ d₁ = 75 / √3 = 25√3 m.Step 3: Using tan(30°) = 1/√3: 1/√3 = 75 / d₂ ⇒ d₂ = 75√3 m.Step 4: Distance between the two men = d₂ − d₁ = 75√3 − 25√3 = 50√3 m.

Verification / Alternative check:
If 50√3 ≈ 50 * 1.732 = 86.6 m, then one man is at about 43.3 m from the pillar and the other at about 129.9 m from the pillar, a reasonable configuration given the two angles 60° (steeper) and 30° (shallower). This matches our trigonometric expectations.

Why Other Options Are Wrong:
Distances like 25√3 m or 75√3 m represent individual distances from the pillar, not the separation between the men. Purely numeric values such as 75 m or 100 m do not satisfy the tangent-based relations for both angles simultaneously.

Common Pitfalls:
Some learners incorrectly add instead of subtracting distances or swap the distances corresponding to 30° and 60°. Remember: a larger angle of elevation means a shorter horizontal distance to the object.

Final Answer:
The distance between the two men is 50√3 m.