From the bank of a river, a person sees that the angle subtended by a tree on the opposite bank is 60°. After walking 40 m directly away from the bank (backwards), the angle becomes 30°. Find the breadth (width) of the river (in metres).

Introduction / Context:
Angle-of-elevation style geometry with two observations at different horizontal distances lets us solve for the unknown width. Here, “angle subtended by the tree” is effectively the angle of elevation to its top from ground level (base angle ≈ 0).

Given Data / Assumptions:

• Initial angle = 60° at distance x (river breadth) from the tree.
• After moving 40 m away, angle = 30° at distance x + 40.
• Tree height = h (unknown, cancels).

Concept / Approach:
tan θ = opposite / adjacent = h / horizontal_distance. Write two equations and eliminate h.

Step-by-Step Solution:

Verification / Alternative check:
Plug back: h = 20√3; second position: tan 30° = (20√3)/(60) = 1/√3 ✔.

Why Other Options Are Wrong:
30, 40, 60, 80 m fail the tangent relations simultaneously for both angles.

Common Pitfalls:
Using sin instead of tan; forgetting the second horizontal distance is x + 40 (farther away).

Final Answer:
20 m