From a point 30 m away from the foot of a flag post, the angles of elevation of the bottom and the top of a flag on the post are 30° and 45° respectively. Taking √3 = 1.732, what is the height of the flag alone?

Introduction / Context:
This problem features a flag mounted on a flag post. From a point on the ground 30 m away, the angles of elevation to the bottom and top of the flag are given. By modelling two right triangles that share the same base distance, we can find the heights of the bottom and top of the flag above the ground and then subtract to get the height of the flag itself.

Given Data / Assumptions:

• Horizontal distance from the observer to the foot of the flag post = 30 m.
• Angle of elevation to the bottom of the flag = 30 degrees.
• Angle of elevation to the top of the flag = 45 degrees.
• The post is vertical and the ground is horizontal.
• Take √3 ≈ 1.732.

Concept / Approach:
The line of sight to the bottom of the flag and the line of sight to the top form two right angled triangles with the same base (30 m). Let the height of the bottom of the flag above ground be h1 and the height of the top be h2. Using tan 30 and tan 45, we can find h1 and h2. The height of the flag alone is h2 − h1. We then approximate using √3 ≈ 1.732 to obtain a decimal answer.

Step-by-Step Solution:
Let h1 be the height of the bottom of the flag. tan 30 = h1 / 30 ⇒ 1 / √3 = h1 / 30 ⇒ h1 = 30 / √3 = 10√3 m. Let h2 be the height of the top of the flag. tan 45 = h2 / 30 ⇒ 1 = h2 / 30 ⇒ h2 = 30 m. Height of the flag alone = h2 − h1 = 30 − 10√3 m. Using √3 ≈ 1.732, 10√3 ≈ 17.32 m. Therefore, height of the flag ≈ 30 − 17.32 = 12.68 m.

Verification / Alternative check:
Check each height separately: h1 ≈ 17.32 m and h2 = 30 m. For the bottom, tan 30 ≈ 0.577 and h1 / 30 ≈ 17.32 / 30 ≈ 0.577, matching tan 30. For the top, tan 45 = 1 and h2 / 30 = 30 / 30 = 1, matching tan 45. The difference 12.68 m therefore reliably represents the vertical height occupied by the flag itself.

Why Other Options Are Wrong:

• 12√3 m: This is about 20.78 m, which is too large given that the whole post plus flag height seen at 45 degrees is only 30 m.
• 15 m: This is a rough guess and does not match 30 − 10√3 exactly.
• 14.32 m: This would require different numeric relationships and does not stem from the correct tan values.
• 10√3 m: This corresponds to the height of the bottom of the flag, not to the height of the flag itself.

Common Pitfalls:
A common error is to treat 30 m as the height or to forget to subtract the two heights, reporting 30 m or 10√3 m directly. Another pitfall is reversing which angle corresponds to which height or mixing up tan 30 and tan 45. Careful reading of the question and a simple diagram usually prevent such confusion.

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