On level ground there is a vertical tower with a flagpole mounted on its top. From a point 9 m away from the foot of the tower along the ground, the angle of elevation of the top of the tower (bottom of the flagpole) is 30°, and the angle of elevation of the top of the flagpole is 60°. What is the height (in metres) of the flagpole alone?

Introduction:
This problem involves two angles of elevation observed from the same point to two different heights on the same vertical line: the top of a tower and the top of a flagpole mounted on that tower. By using basic trigonometry, we can separately determine the height of the tower and the extra height due to the flagpole, then subtract to find the flagpole's height.

Given Data / Assumptions:

• Distance from observation point to foot of the tower = 9 m.
• Angle of elevation to the top of the tower (bottom of flagpole) = 30°.
• Angle of elevation to the top of the flagpole = 60°.
• The tower and flagpole together form a straight vertical line.
• The ground is horizontal and the tower is vertical.

Concept / Approach:
Let H be the height of the tower and (H + h) be the total height including the flagpole, where h is the height of the flagpole. Using tangent for right-angled triangles from the observation point:
tan 30° = H / 9.tan 60° = (H + h) / 9.We use tan 30° = 1/√3 and tan 60° = √3 to solve for H and h.

Step-by-Step Solution:
Step 1: From tan 30° = H/9: 1/√3 = H / 9 ⇒ H = 9 / √3.Step 2: Rationalise: H = 9 / √3 = 9√3 / 3 = 3√3 m.Step 3: From tan 60° = (H + h)/9: √3 = (H + h)/9 ⇒ H + h = 9√3.Step 4: Substitute H = 3√3: 3√3 + h = 9√3.Step 5: Solve for h: h = 9√3 − 3√3 = 6√3 m.

Verification / Alternative check:
If the flagpole height is 6√3 m and the tower is 3√3 m, the total height is 9√3 m. Checking the tangents: for 30°, H/9 = (3√3)/9 = √3/3 = 1/√3, correct; for 60°, (H + h)/9 = (9√3)/9 = √3, also correct.

Why Other Options Are Wrong:
Options such as 5√3 m or 4√3 m do not lead to the correct total height and so fail one or both tangent conditions. 6√2 m and 6√5 m are not related to standard √3-based tangent values for 30° and 60° and are therefore inconsistent with the trigonometric setup.

Common Pitfalls:
Learners sometimes forget that the 60° angle corresponds to the combined height of tower plus flagpole, not just the flagpole. Others mix up tan 30° and tan 60°, or mistakenly use sine instead of tangent. Always clearly define the heights for each angle before writing the equations.

Final Answer:
The height of the flagpole is 6√3 m.