From the top of a 60 m high vertical tower, the angles of depression of the top and bottom of a vertical pole standing on the same level ground are 45° and 60°, respectively. What is the height (in metres) of the pole?

Introduction:
This question involves angles of depression from the top of a tower to both the top and bottom of a pole. The geometry forms two right-angled triangles sharing the same horizontal distance between the tower and the pole. Using tangent relations for the angles of depression, we can determine both the horizontal distance and the height of the pole.

Given Data / Assumptions:

• Height of the tower = 60 m.
• Angle of depression to the top of the pole = 45°.
• Angle of depression to the bottom (foot) of the pole = 60°.
• The tower and the pole stand on the same horizontal ground.
• Both are vertical, and the horizontal distance between them is the same for both lines of sight.

Concept / Approach:
Angles of depression from the top of the tower equal the corresponding angles of elevation from the ground. Let the horizontal distance between the tower and the pole be d, and the height of the pole be h. Then:
From the top to the foot: tan 60° = (60 − 0) / d = 60 / d.From the top to the top of the pole: tan 45° = (60 − h) / d.Using tan 60° = √3 and tan 45° = 1, we can solve for d and h.

Step-by-Step Solution:
Step 1: From the foot of the pole: tan 60° = 60 / d ⇒ √3 = 60 / d ⇒ d = 60 / √3 = 20√3 m.Step 2: From the top of the pole: tan 45° = (60 − h) / d ⇒ 1 = (60 − h) / d ⇒ 60 − h = d.Step 3: Substitute d = 20√3 into 60 − h = 20√3.Step 4: Rearrange to find h: h = 60 − 20√3.Step 5: Factor out 20: h = 20(3 − √3) m.

Verification / Alternative check:
We can check numerically using √3 ≈ 1.732: h ≈ 20(3 − 1.732) = 20 * 1.268 ≈ 25.36 m. With d ≈ 20√3 ≈ 34.64 m, tan 60° ≈ 60/34.64 ≈ 1.732, correct, and tan 45° ≈ (60 − 25.36)/34.64 ≈ 34.64/34.64 ≈ 1, also correct.

Why Other Options Are Wrong:
20(√3 + 1) m and 20(3√3) m are much larger than the tower height or inconsistent with the tangent equations. 60 − 20√3 m is algebraically the same as 20(3 − √3) m, but the fully factored form is clearer in this context; any expression not equivalent to this value will not satisfy both angle conditions.

Common Pitfalls:
Learners often mix up which vertical difference to use for which angle or forget that the vertical drop to the top of the pole is (60 − h), not h. Another common mistake is to use distances with the wrong angle (e.g., using tan 45° for the foot). Drawing both triangles clearly and labeling heights and distances carefully helps avoid confusion.

Final Answer:
The height of the pole is 20(3 − √3) m.