Two men stand on opposite sides of a tower and measure the angles of elevation of the top of the tower as 30° and 45° respectively. If the height of the tower is 50 m (take √3 = 1.73), what is the distance between the two men?

Introduction / Context:
In this problem, two observers stand on opposite sides of a tower and view its top under different angles of elevation. You are given the height of the tower and must find the distance between the two observers. This is a direct application of the tangent function for each right triangle formed with the tower height as the common vertical side.

Given Data / Assumptions:

• Height of the tower, h = 50 m.
• Angle of elevation from the first man = 30 degrees.
• Angle of elevation from the second man = 45 degrees.
• The two men are on opposite sides of the tower on the same straight line.
• The tower is vertical and the ground is horizontal.
• Approximate value: √3 ≈ 1.73.

Concept / Approach:
We form two right triangles with the tower height as the common opposite side and different adjacent sides representing the distances from each man to the tower. Using tan θ = opposite / adjacent, we find each distance. Because the men are on opposite sides, the distance between them is the sum of these two distances. Finally, we substitute √3 ≈ 1.73 to get a decimal value that matches one of the options.

Step-by-Step Solution:
Let d1 be the distance of the man who sees the top at 30 degrees. tan 30 = 50 / d1 ⇒ 1 / √3 = 50 / d1 ⇒ d1 = 50√3 m. Let d2 be the distance of the man who sees the top at 45 degrees. tan 45 = 50 / d2 ⇒ 1 = 50 / d2 ⇒ d2 = 50 m. Distance between the two men = d1 + d2 = 50√3 + 50. Using √3 ≈ 1.73, 50√3 ≈ 50 × 1.73 = 86.5 m. Thus distance ≈ 86.5 + 50 = 136.5 m.

Verification / Alternative check:
Check each triangle separately. For the 30 degree observer: tan 30 ≈ 0.577, and 50 / 0.577 ≈ 86.6 which matches 50√3 ≈ 86.5. For the 45 degree observer: tan 45 = 1, so base = height = 50 m, which is consistent. Adding 86.5 and 50 gives 136.5 m, confirming our result is accurate and consistent with all given data.

Why Other Options Are Wrong:

• 50√3 m: This is only the distance of one man from the tower, not the distance between the two men.
• 100√3 m: This would correspond to each man being 50√3 m away on the same side, which contradicts the 45 degree angle for one observer.
• 135.5 m: This is a near miss but does not match the exact computation 50√3 + 50 with √3 = 1.73.
• 120 m: A rough guess with no basis in the correct trigonometric formulae.

Common Pitfalls:
Learners often subtract the distances instead of adding them because they misinterpret the phrase "opposite sides." It is also easy to confuse tan 30 and tan 60 or to use an inaccurate approximation for √3 without checking. Drawing a diagram and labelling each distance clearly helps avoid such confusion.

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