Height & Distance – Finding pole height from two elevation observations The top of a 15 m tower subtends a 60° angle at the bottom of a nearby electric pole, and a 30° angle at the top of that pole. What is the height of the electric pole?

Introduction / Context:
Two angles of elevation to the same top point (here, the top of a 15 m tower) taken from two different heights (pole bottom and pole top) allow us to determine the unknown pole height using tan relations along a common horizontal line.

Given Data / Assumptions:

• Tower height = 15 m.
• Angle from pole bottom to tower top = 60°.
• Angle from pole top to tower top = 30°.
• Let horizontal separation between tower base and pole base be x > 0.
• Let pole height be h.

Concept / Approach:
From pole bottom: tan 60° = 15/x ⇒ x = 15/√3 = 5√3. From the pole top, vertical difference to tower top is (15 − h), and tan 30° = (15 − h)/x = 1/√3. Solve for h.

Step-by-Step Solution:

Verification / Alternative check:
Check the first observation: tan 60° = 15/(5√3) = √3, valid. Second: tan 30° = (15 − 10)/(5√3) = 5/(5√3) = 1/√3, valid.

Why Other Options Are Wrong:
5 m, 8 m, 12 m give inconsistent tan values for one or both observations.

Common Pitfalls:
Swapping which angle corresponds to which observation point, or forgetting that the second observation sees only the difference in heights.

Final Answer:
10 m