A man standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank is 60°. When he walks 36 m directly away from the river along a straight line perpendicular to the bank, the angle of elevation of the top of the same tree becomes 30°. What is the breadth (in metres) of the river?

Introduction:
This is a classic river-width problem using trigonometry. The observer changes position along a line perpendicular to the bank and sees the angle of elevation of the top of a tree change. From this, we can set up two tangent relations and solve for the distance from the tree to the bank, which is the river's breadth.

Given Data / Assumptions:

• The man initially stands at point A on the bank.
• Angle of elevation of the tree top from A = 60°.
• He then walks 36 m back to point B, directly away from the river.
• Angle of elevation from B = 30°.
• The tree stands vertically on the opposite bank, and the ground is level.

Concept / Approach:
Let x be the breadth of the river (distance from A to the foot of the tree) and h be the height of the tree. From trigonometry in right-angled triangles, we have:
tan 60° = h / x.tan 30° = h / (x + 36).Substituting tan 60° = √3 and tan 30° = 1/√3, we can solve these two equations to find x.

Step-by-Step Solution:
Step 1: From the first position: tan 60° = h/x ⇒ √3 = h/x ⇒ h = √3 * x.Step 2: From the second position: tan 30° = h/(x + 36) ⇒ 1/√3 = h/(x + 36) ⇒ h = (x + 36)/√3.Step 3: Equate the two expressions for h: √3 x = (x + 36)/√3.Step 4: Multiply both sides by √3 to clear the denominator: 3x = x + 36.Step 5: Rearrange: 3x − x = 36 ⇒ 2x = 36 ⇒ x = 18 m.

Verification / Alternative check:
Using x = 18, compute h from h = √3 x: h = 18√3. From the farther point, distance is x + 36 = 54 m. Check tan 30°: h/(x + 36) = (18√3)/54 = √3/3 = 1/√3, which matches tan 30°. Both angles are consistent with x = 18 m as the river width.

Why Other Options Are Wrong:
Values like 15 m, 16 m, 11 m, or 20 m fail to satisfy both trigonometric conditions simultaneously. When used in the equations, they produce conflicting tree heights h, indicating they are not correct solutions.

Common Pitfalls:
Students sometimes treat 36 m as being added on the tree side instead of the observer side or misapply tangent values (interchanging √3 with 1/√3). It is also easy to forget that the second distance is x + 36, not x − 36. Drawing a clear diagram helps avoid these mistakes.

Final Answer:
The breadth of the river is 18 m.