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

Introduction:
This question combines an angle of elevation and an angle of depression from the top of a building to find the height of a nearby tower. By using right-angled triangles and the tangent function, we can find the horizontal distance between the building and the tower and then use it to compute the tower’s total height.

Given Data / Assumptions:

• Height of the building = 20 m.
• Angle of elevation from the building top to the tower top = 60°.
• Angle of depression from the building top to the tower foot = 45°.
• The building and tower stand on the same horizontal ground.
• The tower is vertical and the ground is level.

Concept / Approach:
Let H be the height of the tower and d be the horizontal distance between the building and the tower. From the top of the building:
To the tower foot: tan 45° = (vertical drop) / d = 20 / d.To the tower top: tan 60° = (H − 20) / d.We first find d from the 45° relation and then substitute into the 60° relation to find H.

Step-by-Step Solution:
Step 1: From angle of depression 45° to the foot: tan 45° = 1 = 20 / d ⇒ d = 20 m.Step 2: From angle of elevation 60° to the top: tan 60° = √3 = (H − 20) / d.Step 3: Substitute d = 20: √3 = (H − 20) / 20.Step 4: Rearrange to solve for H: H − 20 = 20√3 ⇒ H = 20√3 + 20.Step 5: Using √3 ≈ 1.732, compute H ≈ 20 * 1.732 + 20 = 34.64 + 20 = 54.64 m.

Verification / Alternative check:
Check back with d = 20 m: For the top, (H − 20)/d ≈ (54.64 − 20)/20 = 34.64/20 ≈ 1.732, consistent with tan 60°. For the foot, 20/d = 20/20 = 1, consistent with tan 45°. Both angles are satisfied accurately.

Why Other Options Are Wrong:
Values like 45.46 m or 45.64 m arise if one mistakenly uses 20 instead of (H − 20) or misuses the 60° relation. 54.46 m is a slight rounding error in the wrong direction and does not match 20(√3 + 1). 40 m ignores the trigonometric relations entirely and is much too small given that the building itself is already 20 m high and the tower must be taller than the building.

Common Pitfalls:
Students often confuse angle of elevation and depression or forget that the angle of depression equals the angle of elevation from the foot of the tower to the building top. Another common error is to use 20 instead of (H − 20) for the vertical side corresponding to the 60° angle. Carefully distinguishing the height difference is crucial.

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