The angles of elevation of the top of a tower 72 metres high, as seen from the top and from the bottom of a nearby building on the same horizontal line, are 30° and 60° respectively. What is the height of the building in metres?

Introduction / Context:
This question involves two vertical structures: a known tower and an unknown building. The angles of elevation of the tower top from the top and bottom of the building are given. Using right-triangle trigonometry, we can set up two equations and solve for both the horizontal distance and the building height, then extract the required height.

Given Data / Assumptions:

• Height of the tower = 72 m.
• Angle of elevation from the top of the building = 30°.
• Angle of elevation from the bottom of the building = 60°.
• The tower and building stand on the same level ground.
• The points (tower base, building base, building top) lie along one straight horizontal line.

Concept / Approach:
Let the building height be h and the horizontal distance between the tower and the building be x. From the bottom of the building, the tower top is at height 72 m, giving tan(60°) = 72 / x. From the top of the building, the vertical difference between the two tops is (72 − h), giving tan(30°) = (72 − h) / x. By solving these two equations simultaneously, we find h.

Step-by-Step Solution:
Let h be the building height and x the horizontal distance.From the building bottom: tan(60°) = 72 / x.tan(60°) = √3 ⇒ √3 = 72 / x ⇒ x = 72 / √3 = 24√3.From the building top: tan(30°) = (72 − h) / x.tan(30°) = 1 / √3 ⇒ 1 / √3 = (72 − h) / x.Substitute x = 24√3: 1 / √3 = (72 − h) / (24√3).Cross-multiply: (72 − h) = (24√3) / √3 = 24.Therefore, h = 72 − 24 = 48 m.

Verification / Alternative check:
With h = 48 m and x = 24√3, from the bottom: tan(60°) = 72 / (24√3) = 3 / √3 = √3, correct. From the top: the difference in height is 72 − 48 = 24 m, so tan(30°) = 24 / (24√3) = 1 / √3, also correct. Both triangles check out perfectly.

Why Other Options Are Wrong:
42 m and 36 m: These intermediate values do not satisfy both tangent equations simultaneously.20√3 m and 24√3 m: These are associated with horizontal distances, not the building height itself, although 24√3 does appear in the working as x.

Common Pitfalls:
One common mistake is to assume the building is taller than the tower or misinterpret which height difference to use from the top of the building. Another is mixing up tan(30°) and tan(60°). Carefully identify vertical differences (72 − h) from the top and full height 72 from the bottom to build correct equations.

Final Answer:
The height of the building is 48 m.