A pilot flying at an altitude of 200 m observes two points on opposite banks of a river. The angles of depression of the two points are 45° and 60° respectively. What is the width of the river?

Introduction / Context:
This question describes an aeroplane flying horizontally over a river while a pilot observes two points on opposite banks at different angles of depression. Because both observations are made from the same altitude, we get two right triangles sharing the same vertical height but with different horizontal distances from the point on the ground directly below the aeroplane. The width of the river is the distance between these two ground points.

Given Data / Assumptions:

• Altitude of the aeroplane above the river surface = 200 m.
• Angles of depression to the two opposite banks = 45 degrees and 60 degrees.
• The river banks lie on a straight line on either side of the point vertically below the aeroplane.
• Angles of depression equal the corresponding angles of elevation from the river banks.
• Standard values: tan 45 = 1, tan 60 = sqrt(3).

Concept / Approach:
Let the point vertically below the aeroplane on the water surface be P. Let A and B be the points on the nearer and farther banks respectively, such that angles of depression are 60 degrees to A and 45 degrees to B. Then AP and BP are horizontal distances. For each right triangle, tan θ = height / base. We find AP and BP and then sum them to get the total width AB = AP + BP, because the banks lie on opposite sides of P.

Step-by-Step Solution:
Let AP be the horizontal distance to the nearer bank where the angle of depression is 60 degrees. Then tan 60 = 200 / AP ⇒ sqrt(3) = 200 / AP ⇒ AP = 200 / sqrt(3) m. Let BP be the horizontal distance to the farther bank where the angle of depression is 45 degrees. Then tan 45 = 200 / BP ⇒ 1 = 200 / BP ⇒ BP = 200 m. Width of the river AB = AP + BP = 200 / √3 + 200. Thus, width = (200 + 200/√3) m.

Verification / Alternative check:
Using √3 ≈ 1.732, AP ≈ 200 / 1.732 ≈ 115.47 m and BP = 200 m. So the width is about 315.47 m. Both triangles use the same vertical height 200 m. For AP, tan 60 ≈ 200 / 115.47 ≈ 1.732, which matches sqrt(3). For BP, tan 45 = 200 / 200 = 1, as expected. The sum of the distances matches the derived algebraic expression.

Why Other Options Are Wrong:

• (200 − 200/√3) m: This would be the difference of distances, not the sum, and does not represent the full river width when the plane is above a point between the banks.
• 400√3 m and (400/√3) m: These values are much larger or smaller than what the tan relations provide and would not maintain the correct angle relationships.
• 200√3 m: This corresponds to a single base distance with tan 60, not the combined width of both banks.

Common Pitfalls:
Students may mistakenly subtract distances instead of adding them by misinterpreting the location of the aeroplane relative to the river banks. Another error is to confuse which angle corresponds to the nearer bank and which to the farther bank. Remember that the larger angle of depression corresponds to the closer point (shorter base distance). Drawing a simple diagram is very helpful in avoiding such mistakes.

Final Answer:
The width of the river is (200 + 200/√3) m.