A pilot in an aeroplane is flying at a constant altitude of 200 m above a river. He observes two points lying on opposite banks of the river such that the angles of depression to these two points are 45° and 60° respectively. Assuming the plane is vertically above some point between the two banks, what is the width of the river in metres?

Introduction / Context:
This question involves angles of depression from a fixed altitude to two points on opposite sides of a river. By interpreting the angles as angles of elevation from the ground, we can form two right triangles sharing the same vertical height and find the horizontal distances to each bank. Adding these distances gives the width of the river.

Given Data / Assumptions:

• The aeroplane flies at a constant height h = 200 m above the river.
• Angles of depression to the two points on opposite banks are 45° and 60°.
• The plane is vertically above a point between the two banks on a straight line perpendicular to the river.
• The river banks are assumed straight and parallel.

Concept / Approach:
Angles of depression from the horizontal are equal to the corresponding angles of elevation from the ground. Thus, for each bank we use:
tan(theta) = opposite / adjacent = height / horizontal distanceWe then calculate the horizontal distance from the point directly beneath the plane to each bank and sum them to get the river width.

Step-by-Step Solution:
Let h = 200 m.For bank A with angle 45°: tan(45°) = h / d1 ⇒ 1 = 200 / d1 ⇒ d1 = 200 m.For bank B with angle 60°: tan(60°) = h / d2 ⇒ √3 = 200 / d2.So, d2 = 200 / √3 m.The two points lie on opposite sides, so the river width W = d1 + d2.Therefore, W = 200 + 200 / √3 m.

Verification / Alternative check:
We may approximate √3 ≈ 1.732. Then 200 / √3 ≈ 200 / 1.732 ≈ 115.47 m. The total width becomes approximately 200 + 115.47 = 315.47 m. Both triangles are consistent with the altitude of 200 m and the given angles using tan(45°) and tan(60°), confirming our formula.

Why Other Options Are Wrong:
(200 - 200/√3) m: This would correspond to an unrealistic subtraction of distances; width must be the sum of distances to opposite banks.400 √3 m and 400/√3 m: These are much larger or smaller than the correct width and do not match the individual triangle distances.200√3 m: Roughly 346 m, which would come from using tan(60°) with a single triangle rather than combining two banks properly.

Common Pitfalls:
Students commonly misinterpret “opposite banks” and subtract instead of add the distances. Another frequent mistake is reversing the roles of 45° and 60°, or misusing tan(60°) as 1 / √3. Remember that angles of depression equal angles of elevation, and that width must be the total separation between the two banks.

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