A helicopter is flying at an altitude of 1500 m above the sea. It observes two ships sailing in the same straight line and in the same direction. The angles of depression of the two ships from the helicopter are 60° and 30° respectively. What is the distance between the two ships in metres?

Introduction / Context:
This problem involves angles of depression from a helicopter to two ships at sea level. Since both ships are in the same direction from the helicopter and the helicopter's height is fixed, we can use trigonometry separately for each angle of depression and then find the distance between the ships along the sea surface.

Given Data / Assumptions:

• Helicopter altitude h = 1500 m.
• Angles of depression to the nearer and farther ships are 60° and 30° respectively.
• The ships lie in the same straight line beneath the helicopter and in the same direction.
• Sea level is taken as a horizontal line, creating right triangles with the helicopter's vertical altitude.

Concept / Approach:
An angle of depression from the helicopter equals the angle of elevation from the ship. For each ship, we use:
tan(theta) = opposite / adjacent = height / horizontal distanceWe calculate the horizontal distances d1 and d2 for 60° and 30°, and then take the difference d2 − d1 to find the separation between the ships.

Step-by-Step Solution:
Let h = 1500 m.For the nearer ship with angle 60°: tan(60°) = h / d1 ⇒ √3 = 1500 / d1.Thus, d1 = 1500 / √3 = 500√3 m.For the farther ship with angle 30°: tan(30°) = h / d2 ⇒ 1 / √3 = 1500 / d2.So, d2 = 1500√3 m.Distance between the two ships = d2 − d1 = 1500√3 − 500√3 = 1000√3 m.

Verification / Alternative check:
Approximating √3 ≈ 1.732, we get:
d1 ≈ 500 * 1.732 ≈ 866 m,d2 ≈ 1500 * 1.732 ≈ 2598 m.Difference ≈ 2598 − 866 ≈ 1732 m, which equals 1000 * 1.732, confirming the expression 1000√3 m.

Why Other Options Are Wrong:
1000/√3 m and 500/√3 m: Too small; they come from inverting the tangent relationships incorrectly.500√3 m: This is the distance from the helicopter's projection to the nearer ship, not the distance between the two ships.1500√3 m: This is the distance to the farther ship from the projection, again not the separation between the two ships.

Common Pitfalls:
Students sometimes add the distances instead of subtracting because they misinterpret the positions, or they mix up which angle corresponds to which distance. Another frequent mistake is to confuse tan(30°) with tan(60°) or to use 1 / tan instead of tan itself. Drawing a clear diagram helps avoid these issues.

Final Answer:
The distance between the two ships is 1000√3 m.