From a point 30 metres away from the foot of a flag post, the angles of elevation of the top and the bottom of a flag fixed on the post are 45° and 30° respectively. Assuming the post is vertical and the ground is level, what is the height of the flag itself (in metres)? Take √3 = 1.732 where needed.

Introduction / Context:
This problem distinguishes between the total height of a flag post and the height of the flag attached to it. Two different angles of elevation, to the top and the bottom of the flag, are observed from the same point, allowing us to calculate both heights and then subtract to obtain the flag's height.

Given Data / Assumptions:

• Observer is 30 m horizontally from the base of the flag post.
• Angle of elevation to the top of the flag = 45°.
• Angle of elevation to the bottom of the flag = 30°.
• The post is vertical, and the ground is horizontal.
• We must find only the height of the flag (difference between these two vertical heights).

Concept / Approach:
Both lines of sight form right triangles with the same base of 30 m. The heights corresponding to 45° and 30° angles of elevation can be obtained via:
tan(theta) = opposite / adjacentOnce we find the vertical height to the top and to the bottom of the flag, their difference equals the flag height.

Step-by-Step Solution:
Base distance d = 30 m.Height to top of flag, H_top satisfies: tan(45°) = H_top / 30.tan(45°) = 1 ⇒ H_top = 30 m.Height to bottom of flag, H_bottom satisfies: tan(30°) = H_bottom / 30.tan(30°) = 1 / √3 ⇒ H_bottom = 30 * (1 / √3) = 30 / √3.Rationalizing: 30 / √3 = 10√3.Using √3 ≈ 1.732 ⇒ H_bottom ≈ 10 * 1.732 = 17.32 m.Height of the flag = H_top − H_bottom ≈ 30 − 17.32 = 12.68 m.

Verification / Alternative check:
If the flag is approximately 12.68 m high, then adding its bottom height (~17.32 m) indeed gives the top at 30 m. Reversing the calculations back through tangent confirms that these heights correspond to angles of 30° and 45° respectively, verifying consistency.

Why Other Options Are Wrong:
12√3 m (~20.78 m): This is larger than the total height difference and would place the top higher than 30 m.15 m and 14.32 m: Plausible but not matching the exact trigonometric calculations.10√3 m (~17.32 m): This is the height to the bottom of the flag, not the flag's height itself.

Common Pitfalls:
Many learners mistakenly compute the total height of the post or take only one of the heights instead of subtracting. Misapplying tan(30°) as √3 instead of 1 / √3 is another frequent source of error. Keeping track of which angle corresponds to which point (top or bottom of the flag) is critical.

Final Answer:
The height of the flag is approximately 12.68 m.