A flagpole is placed at the top of a tower. The angles of elevation of the bottom and top of the flagpole from…

Analyze the question: From a point 10 m from the base of the tower, the angle of elevation to the bottom of the flagpole (the top of the tower) is 40°, and to the top of the flagpole it is 50°. We need the flagpole's length, which is the difference between the two heights.

Concept: In a right-angled triangle, tan θ = opposite/adjacent. Here the adjacent side is the 10 m horizontal distance, so height = 10 tan θ.

Solution:

• Height of tower: h₁ = 10 tan 40° = 10 × 0.8391 = 8.391 m
• Height to the top of the flagpole: h₂ = 10 tan 50° = 10 × 1.1918 = 11.918 m

Length of flagpole = h₂ − h₁ = 11.918 − 8.391 = 3.527 m ≈ 3.5 m

Note that C (8.4 m) is the tower's height and D (11.9 m) is the total height. These are the common wrong answers when you forget to subtract.

Tip: In "tower plus flagpole" questions, compute both heights using tan, then subtract. Always check whether the question asks for the part or the total.