From the top of a 20 m high building, the angle of elevation of the top of a tower is 60° and the angle of depression of the foot of the tower is 45°. What is the height of the tower? (Take √3 = 1.732)

Introduction / Context:
This question combines angles of elevation and depression to relate the heights of a building and a tower standing on the same horizontal ground. From the top of a known height building, you observe both the top and the foot of a tower at different angles. The objective is to use trigonometry to find the total height of the tower by first obtaining the horizontal distance between the two structures and then using that distance with the angle of elevation.

Given Data / Assumptions:

• Height of the building = 20 m.
• Angle of depression to the foot of the tower = 45 degrees.
• Angle of elevation to the top of the tower = 60 degrees.
• The building and tower stand on the same level ground.
• Standard values: tan 45 = 1, tan 60 = sqrt(3) (use sqrt(3) ≈ 1.732).

Concept / Approach:
First, use the angle of depression to find the horizontal distance between the building and the tower. An angle of depression from the horizontal at the top equals the angle of elevation from the foot to the top of the building. With the height of the building and tan 45, we can find the horizontal distance. Then apply the angle of elevation (60 degrees) from the building top to the tower top to get the extra height above the building. Adding this extra height to 20 m gives the total tower height.

Step-by-Step Solution:
Let the horizontal distance between the building and tower be d. From the 45 degree angle of depression, tan 45 = opposite / adjacent = 20 / d. Since tan 45 = 1, we get 1 = 20 / d, so d = 20 m. Let the height of the tower be H. From the 60 degree angle of elevation: tan 60 = (H - 20) / d. So sqrt(3) = (H - 20) / 20, which implies H - 20 = 20√3. Using √3 ≈ 1.732, H - 20 ≈ 20 × 1.732 = 34.64. Therefore, H ≈ 20 + 34.64 = 54.64 m.

Verification / Alternative check:
A quick check: With H ≈ 54.64 m, the height difference above the building is about 34.64 m. The ratio (H - 20) / d ≈ 34.64 / 20 ≈ 1.732, which matches tan 60. Also, the distance d = 20 m and building height 20 m give tan 45 = 20 / 20 = 1, which is correct. This confirms that the computed height is consistent with both angles given.

Why Other Options Are Wrong:

• 45.46 m and 45.64 m: These heights are too small and would produce a much smaller angle than 60 degrees from the building top.
• 54.46 m: Close but slightly off; it does not match the exact use of √3 = 1.732 in the calculation.
• 50.00 m: This ignores the 60 degree elevation relationships and is not supported by the trigonometric ratios.

Common Pitfalls:
Students sometimes confuse which segment corresponds to the opposite or adjacent side for each angle. Another common error is to forget that the tower height above the building is H minus 20, not H. Misinterpreting the angle of depression as something different from the corresponding angle of elevation at ground level can also lead to mistakes. A clear labelled diagram is very helpful.

Final Answer:
The height of the tower is approximately 54.64 m.