Two persons stand on opposite sides of a temple 75 m high and observe the top of the temple at angles of elevation 30° and 60° respectively. What is the distance between the two persons?

Introduction / Context:
This question involves two observers standing on opposite sides of a tall temple and looking at its top at different angles of elevation. The situation forms two separate right triangles that share the same vertical height (the height of the temple). The aim is to use basic trigonometry to find each observer's horizontal distance from the temple and then add them to get the distance between the two people.

Given Data / Assumptions:

• Height of the temple, h = 75 m.
• First person sees the top at an angle of elevation θ1 = 30 degrees.
• Second person sees the top at an angle of elevation θ2 = 60 degrees.
• The persons are on opposite sides of the temple on level ground.
• The temple is vertical, forming right angled triangles with the ground.
• Standard values: tan 30 = 1 / sqrt(3), tan 60 = sqrt(3).

Concept / Approach:
For each observer, we set up a right triangle with the temple height as the opposite side and the horizontal distance from the temple as the adjacent side. Using tan θ = opposite / adjacent, we can find each horizontal distance. Since the two observers are on opposite sides of the temple in a straight line, the distance between them is the sum of these two distances.

Step-by-Step Solution:
Let x be the distance of the observer with 60 degree elevation from the temple. Then tan 60 = 75 / x, so sqrt(3) = 75 / x, giving x = 75 / sqrt(3) = 25√3 m. Let y be the distance of the observer with 30 degree elevation from the temple. Then tan 30 = 75 / y, so 1 / sqrt(3) = 75 / y, giving y = 75√3 m. Distance between the two persons = x + y = 25√3 + 75√3 = 100√3 m. Using sqrt(3) ≈ 1.732, 100√3 ≈ 173.2 m.

Verification / Alternative check:
A quick approximate check: 25√3 is about 43.3 m and 75√3 is about 129.9 m. Their sum is about 173.2 m, which matches the given option exactly. The geometry makes sense: one observer closer to the temple sees a larger angle of elevation (60 degrees), while the more distant observer sees a smaller angle (30 degrees).

Why Other Options Are Wrong:

• 100 m: This ignores the trigonometric ratios and underestimates the total distance.
• 157.7 m: This is near but does not match the exact value 100√3 ≈ 173.2.
• 273.2 m: This is too large and would correspond to much greater separation than the actual tan values produce.
• 150 m: A rounded guess with no proper trigonometric basis in this context.

Common Pitfalls:
Students may mistakenly subtract the distances instead of adding them, forgetting that the observers are on opposite sides. Another frequent mistake is confusing tan 30 and tan 60, which swaps the distances and leads to incorrect totals. Always draw a simple diagram marking angles and sides, and remember that higher angles of elevation correspond to shorter horizontal distances for a fixed height.

Final Answer:
The distance between the two persons is 173.2 m (that is, 100√3 m approximately).