A man stands on the bank of a river and 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 bank, the angle of elevation of the top of the same tree becomes 30°. What is the breadth (width) of the river?

Introduction / Context:
This question is a classic example of using trigonometric ratios with two different observation points. A man observes the top of a tree across a river from two positions along a straight line perpendicular to the river bank. By measuring how the angle of elevation changes when he moves a known distance farther away, we can determine the unknown width of the river without directly measuring it. The tree is assumed to be vertical, and the ground on the man's side of the river is taken as level.

Given Data / Assumptions:

- The tree stands vertically on the opposite bank of the river. - The man initially stands at the river bank. - Angle of elevation of the top of the tree at the bank is 60°. - He then walks 36 m away from the bank in a straight line perpendicular to the river. - From the new point, the angle of elevation of the top of the tree is 30°. - The river banks are assumed parallel and the ground is level.

Concept / Approach:
Let the breadth of the river be w metres and the height of the tree be h metres. When the man is at the river bank, the horizontal distance from him to the tree is w. When he moves 36 m away, the distance becomes w + 36. The angle of elevation in each case gives us a tangent relation: tan 60° = h / w and tan 30° = h / (w + 36). Because both expressions equal the same tree height h, we can equate them and solve for w. This method avoids the need to calculate h explicitly until after we have found w, which is the required breadth of the river.

Step-by-Step Solution:
Step 1: Let w be the width of the river and h be the height of the tree. Step 2: At the river bank, the angle of elevation is 60°, so tan 60° = h / w. Step 3: Using tan 60° = √3, we get h / w = √3, so h = w√3. Step 4: After walking 36 m away from the bank, the horizontal distance becomes w + 36, and the angle of elevation is 30°. Step 5: For this position, tan 30° = h / (w + 36). Using tan 30° = 1 / √3, we have 1 / √3 = h / (w + 36). Step 6: Substitute h = w√3 into the second equation to get 1 / √3 = (w√3) / (w + 36). Step 7: Multiply both sides by (w + 36)√3 to get w + 36 = 3w. Step 8: Rearranging gives 3w - w = 36, so 2w = 36 and w = 18. Step 9: Therefore the breadth of the river is 18 m.

Verification / Alternative check:
We can verify by computing the height h. From h = w√3 with w = 18, we get h = 18√3 m. At the bank, tan 60° = h / w = (18√3) / 18 = √3, which is correct. At the new point, the distance is w + 36 = 18 + 36 = 54 m. The tangent of the new angle is h / (w + 36) = (18√3) / 54 = √3 / 3 = 1 / √3, which is the correct value for tan 30°. Both angles match the problem statement, confirming that w = 18 m is correct.

Why Other Options Are Wrong:
Option A (15 m) and option C (16 m) do not satisfy both tangent equations when substituted, so the angles will not come out as exactly 60° and 30°. Option D (11 m) is even smaller and clearly inconsistent with the geometry of the movement of 36 m away from the bank. Only 18 m produces a consistent pair of distances and heights that give the given angles of elevation.

Common Pitfalls:
One common mistake is to confuse which distance corresponds to which angle, for example taking w + 36 for the 60° case and w for the 30° case. Another error is to subtract 36 from w in the second position instead of adding, which does not match the described motion. Some students also incorrectly try to solve for h first in one equation and plug into the other without careful algebra, leading to sign errors. Drawing a clear diagram with both positions labelled and writing separate tangent equations helps avoid these issues.

Final Answer:
The breadth of the river is 18 m, which corresponds to option B.