On level ground, there stands a vertical tower with a flagpole fixed on its top. From a point on the ground 9 m away from the foot of the tower, the angles of elevation of the top and the bottom of the flagpole are 60° and 30° respectively. What is the height of the flagpole?

Introduction / Context:
This problem is a standard height and distance question based on right angled triangles and trigonometric ratios. A vertical tower has a flagpole fixed on its top, and an observer stands on level ground at a known horizontal distance from the foot of the tower. From that point, the observer measures two angles of elevation: one to the bottom of the flagpole (the top of the tower itself), and the other to the top of the flagpole. Using these two angles and the known horizontal distance, we can set up two tangent relations and then find the unknown height of the flagpole alone, without needing to know the total height of the tower in advance.

Given Data / Assumptions:

- The tower is vertical and stands on level ground. - A flagpole is fixed vertically on the top of the tower. - The observer is 9 m away from the foot of the tower along level ground. - Angle of elevation to the bottom of the flagpole (top of tower) is 30°. - Angle of elevation to the top of the flagpole is 60°. - We assume standard trigonometric values: tan 30° = 1 / √3 and tan 60° = √3.

Concept / Approach:
The key idea is to treat the situation as two right angled triangles that share the same horizontal base (9 m) but have different vertical heights. Let the height of the tower be H and the height of the flagpole be h. Then the total height from ground to the top of the flagpole is H + h. The angle of elevation to the top of the tower uses H in the tangent ratio, while the angle of elevation to the top of the flagpole uses H + h. By writing two tangent equations in terms of H and H + h and then subtracting, we can eliminate H and solve directly for h, the height of the flagpole.

Step-by-Step Solution:
Step 1: Let H be the height of the tower (up to the bottom of the flagpole) and h be the height of the flagpole itself. Step 2: From the point on the ground, the angle of elevation to the bottom of the flagpole is 30°, so tan 30° = H / 9. Step 3: Using tan 30° = 1 / √3, we get H / 9 = 1 / √3, so H = 9 / √3 = 3√3. Step 4: The angle of elevation to the top of the flagpole is 60°, so tan 60° = (H + h) / 9. Step 5: Using tan 60° = √3, we have (H + h) / 9 = √3, so H + h = 9√3. Step 6: Substitute H = 3√3 into H + h = 9√3 to get 3√3 + h = 9√3. Step 7: Rearranging gives h = 9√3 - 3√3 = 6√3. Step 8: Therefore the height of the flagpole is 6√3 m, which is approximately 10.39 m if we need a decimal value.

Verification / Alternative check:
We can verify the result by checking both triangles numerically. If H = 3√3 m and h = 6√3 m, then H + h = 9√3 m. Using the distance 9 m, tan 30° = H / 9 = (3√3) / 9 = √3 / 3, which matches the standard value of tan 30°. For the top of the flagpole, tan 60° = (H + h) / 9 = (9√3) / 9 = √3, which matches tan 60°. Since both trigonometric relations are satisfied exactly, the height 6√3 m is consistent and correct.

Why Other Options Are Wrong:
Option A (5√3 m) gives a total height H + h = 3√3 + 5√3 = 8√3 m, which would not produce tan 60° = √3 with a base of 9 m. Option C (6√2 m) does not match the standard tangent values for 30° and 60° when substituted back into the equations. Option D (6√5 m) similarly fails the tangent checks and does not satisfy both angle conditions simultaneously. Only 6√3 m preserves the exact trigonometric relationships for both given angles from the same observation point.

Common Pitfalls:
A common mistake is to treat the two given angles as if they belonged to two different base distances instead of the same 9 m base. Another frequent error is mixing up which angle corresponds to which height (tower top versus flagpole top), or assuming that the tower height is given directly. Students may also incorrectly use sine instead of tangent for heights over horizontal distances in such problems. Careful diagram drawing, correct identification of opposite and adjacent sides, and writing separate equations for each angle help avoid these errors.

Final Answer:
The height of the flagpole is 6√3 m, which corresponds to option B.