From a point on level ground, the angle of elevation of the top of a building is 45°. After a person walks 100 metres straight towards the building in the same horizontal line, the angle of elevation becomes 60°. What is the height of the building (in metres)?

Introduction / Context:
This question involves a person moving closer to a building, causing the angle of elevation of the top to increase from 45° to 60°. Using trigonometry and the known horizontal movement, we can form equations to find the horizontal distance to the building and hence its height.

Given Data / Assumptions:

• Initial angle of elevation to the top of the building = 45°.
• After walking 100 m towards the building, the angle of elevation = 60°.
• The walking path is along a straight horizontal line directly towards the building.
• The building stands vertically on level ground.
• We are asked to find the building height.

Concept / Approach:
Let x be the initial horizontal distance from the observer to the building and h the building height. Initially, tan(45°) = h / x, giving h = x. After walking 100 m, the new distance is (x − 100), and tan(60°) = h / (x − 100). Substituting h = x into the second equation allows us to solve for x and then find h.

Step-by-Step Solution:
Let h be the height of the building and x the initial distance.From the first position: tan(45°) = h / x ⇒ 1 = h / x ⇒ h = x.From the second position: distance to building = x − 100.tan(60°) = h / (x − 100) ⇒ √3 = h / (x − 100).Substitute h = x: √3 = x / (x − 100).Cross-multiply: √3(x − 100) = x.√3x − 100√3 = x ⇒ √3x − x = 100√3.x(√3 − 1) = 100√3 ⇒ x = 100√3 / (√3 − 1).Multiply numerator and denominator by (√3 + 1):x = 100√3(√3 + 1) / (3 − 1) = 50√3(√3 + 1).Since √3(√3 + 1) = 3 + √3, we get x = 50(3 + √3).But h = x, so height h = 50(3 + √3) m.

Verification / Alternative check:
Approximating √3 ≈ 1.732, h ≈ 50(3 + 1.732) = 50 * 4.732 ≈ 236.6 m. Initially, distance x ≈ 236.6 m gives tan(45°) = h / x ≈ 1. After walking 100 m, new distance ≈ 136.6 m, and h / (x − 100) ≈ 236.6 / 136.6 ≈ 1.732 = √3, which is tan(60°). This confirms consistency.

Why Other Options Are Wrong:
100(√3 + 1) m and 50(√3 + 1) m: These come from partial or incorrect algebraic manipulation and do not satisfy both angle conditions.150 m and 100√3 m: These simple values fail when substituted back into the tangent relations for both 45° and 60°.

Common Pitfalls:
Students often forget that the building height equals the initial distance when the angle is 45°, or they mistakenly use (x + 100) instead of (x − 100) for the second distance. Confusion between tan(45°) and tan(60°), as well as arithmetic errors in handling √3, are also common.

Final Answer:
The height of the building is 50(3 + √3) m.