Two men stand on opposite sides of a vertical tower. One measures the angle of elevation of the top of the tower as 30°, and the other measures the angle of elevation as 45°. If the height of the tower is 50 m, what is the distance between the two men (in metres)? Take √3 = 1.73 where needed.

Introduction / Context:
This question applies trigonometry to a situation where two observers stand on opposite sides of a tower and view its top at different angles of elevation. You must use the tangent ratio to find each observer's distance from the tower and then add them to obtain the separation between the two men.

Given Data / Assumptions:

• The tower is vertical and 50 m high.
• Man A sees the top of the tower at an angle of elevation of 30°.
• Man B, on the opposite side, sees the top at an angle of 45°.
• Both men are assumed to stand on level ground at the same horizontal level as the base of the tower.
• The tower, both observers and the horizontal ground form right triangles.

Concept / Approach:
For each observer, the relationship between the tower height h and horizontal distance d from the tower is:
tan(theta) = h / d ⇒ d = h / tan(theta)We compute distances d1 and d2 for 30° and 45° respectively, then sum them because the observers are on opposite sides of the tower in a straight line.

Step-by-Step Solution:
Height h = 50 m.For the observer at 30°: tan(30°) = 1 / √3.So, d1 = h / tan(30°) = 50 / (1 / √3) = 50√3.Using √3 ≈ 1.73, d1 ≈ 50 * 1.73 = 86.5 m.For the observer at 45°: tan(45°) = 1.So, d2 = h / tan(45°) = 50 / 1 = 50 m.The men are on opposite sides of the tower, so distance between them = d1 + d2.Total distance ≈ 86.5 + 50 = 136.5 m.

Verification / Alternative check:
We can quickly check that the approximate value matches the symbolic sum 50√3 + 50 = 50(√3 + 1). With √3 ≈ 1.73, this becomes 50 * 2.73 = 136.5 m, exactly the value used above. So our computations are consistent.

Why Other Options Are Wrong:
50√3 m (~86.6 m): This is only the distance of the 30° observer from the tower, not the distance between the two men.100√3 m (~173.2 m): Overestimates the distance; comes from mis-adding or doubling 50√3.135.5 m: Close but slightly off; likely due to rounding or arithmetic error.86.6 m: Approximates only one side distance, not the separation.

Common Pitfalls:
Students sometimes subtract the distances instead of adding them because they misinterpret “opposite sides.” Another error is using tan(theta) = d / h instead of h / d. Also, not applying the given approximation √3 = 1.73 carefully can produce minor numerical discrepancies.

Final Answer:
The distance between the two men is approximately 136.5 m.