Two persons stand on opposite sides of a temple that is 75 m high. The angles of elevation of the top of the temple from these two persons are 30° and 60°, respectively. Assuming the ground is level, what is the distance (in metres) between the two persons?

Introduction:
This height and distance problem involves two observers on opposite sides of a temple of known height, each seeing the top under different angles of elevation. By using right-angled triangles and the tangent function, we can determine how far each observer is from the temple and then sum these distances to find the distance between the two persons.

Given Data / Assumptions:

• Height of the temple = 75 m.
• Angle of elevation from person A = 30°.
• Angle of elevation from person B (on the other side) = 60°.
• The ground is horizontal and the temple is vertical.
• The two observers are on the same straight line passing through the base of the temple but on opposite sides.

Concept / Approach:
Let d₁ be the distance of person A (seeing 30°) from the temple and d₂ be the distance of person B (seeing 60°). For each right triangle, we use:
tan θ = height / distance.The total distance between the two persons is d₁ + d₂.

Step-by-Step Solution:
Step 1: For person A (30°): tan 30° = 1/√3 = 75 / d₁ ⇒ d₁ = 75√3.Step 2: For person B (60°): tan 60° = √3 = 75 / d₂ ⇒ d₂ = 75 / √3 = 25√3.Step 3: Distance between the two persons = d₁ + d₂ = 75√3 + 25√3 = 100√3.Step 4: Using √3 ≈ 1.732, compute: 100√3 ≈ 100 * 1.732 ≈ 173.2 m.

Verification / Alternative check:
Check that the distances are reasonable: The observer with the smaller angle (30°) must be farther away (75√3 ≈ 129.9 m), and the observer with the larger angle (60°) must be closer (25√3 ≈ 43.3 m). Adding them gives ≈ 173.2 m, which fits the geometry and the trigonometric expectations.

Why Other Options Are Wrong:
100 m or 200 m do not correspond to 100√3 and do not match the correct addition of 75√3 and 25√3. 157.7 m and 273.2 m are distractors based on incorrect combinations or approximations of √3. Only 173.2 m is equal to 100√3 and satisfies all given conditions.

Common Pitfalls:
A typical error is to subtract the distances instead of adding them, forgetting that the observers are on opposite sides of the temple. Another mistake is to reverse the use of 30° and 60°, leading to inverted distances. Always remember: smaller angle → farther distance; larger angle → closer distance.

Final Answer:
The distance between the two persons is approximately 173.2 m.