From the top and from the bottom of a building, the angles of elevation of the top of a vertical tower 72 metres high are 30° and 60° respectively. What is the height, in metres, of the building?

Introduction / Context:
This question is based on heights and distances, where angles of elevation are used to relate vertical heights and horizontal distances in right angled triangles. We are given a tower of known height and a building of unknown height, and two different angles of elevation of the tower top observed from the bottom and from the top of the building. Using simple trigonometry with tangent ratios, we can form equations and determine the height of the building.

Given Data / Assumptions:

• The height of the vertical tower is 72 metres.
• From the bottom of the building, the angle of elevation of the top of the tower is 60 degrees.
• From the top of the building, the angle of elevation of the top of the tower is 30 degrees.
• The building and the tower stand on the same horizontal ground.
• The horizontal distance between the building and the tower is the same for both observations.

Concept / Approach:
In height and distance problems, the tangent of an angle of elevation is equal to vertical height divided by horizontal distance. Here we use tan 60 degrees and tan 30 degrees. Let the height of the building be h metres and the horizontal distance between the building and the tower be x metres. From the two angles, we obtain two equations in terms of h and x and then solve them simultaneously to find h.

Step-by-Step Solution:
Step 1: From the bottom of the building, tan 60° = 72 / x, so √3 = 72 / x and therefore x = 72 / √3 = 24√3 metres. Step 2: From the top of the building, the vertical difference between the top of the tower and the top of the building is 72 − h. Step 3: The angle of elevation from the top is 30°, so tan 30° = (72 − h) / x. Therefore 1 / √3 = (72 − h) / x. Step 4: Substitute x = 24√3 into the equation: 1 / √3 = (72 − h) / (24√3). Step 5: Multiply both sides by 24√3 to get 24 = 72 − h, so h = 72 − 24 = 48 metres.

Verification / Alternative check:
Check with both angles. With x = 24√3 and h = 48, the vertical difference from bottom is 72, so tan 60° = 72 / (24√3) = 72 / (24 * 1.732) which simplifies to √3, correct. The vertical difference from top is 72 − 48 = 24, so tan 30° = 24 / (24√3) = 1 / √3, correct. Therefore the value h = 48 metres is consistent for both observations.

Why Other Options Are Wrong:
Option 42 would give a vertical difference of 30 metres from the top, which does not produce tan 30° correctly. Option 20√3 and option 24√3 are heights that do not satisfy both angle equations simultaneously. Option 36 also fails when checked with tan 30° and tan 60° conditions. Only 48 metres satisfies both trigonometric relationships exactly.

Common Pitfalls:
A common mistake is to interchange the angles, or to forget that the vertical distance from the top of the building to the top of the tower is 72 minus h and not 72 plus h. Another frequent error is mishandling √3 when simplifying expressions such as 72 / √3. Students sometimes also confuse angle of elevation with angle of depression, although here only angles of elevation are used.

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