Two ships are on opposite sides of a vertical lighthouse. The angles of elevation of the top of the lighthouse from the two ships are 30° and 45°, respectively. If the lighthouse is 100 m high (take √3 ≈ 1.73), what is the distance (in metres) between the two ships?

Introduction:
This is a straightforward height and distance problem involving a lighthouse and two ships on opposite sides of it. By using basic trigonometry, we can find the horizontal distance from each ship to the lighthouse and then add these distances to get the total separation between the ships.

Given Data / Assumptions:

• Height of the lighthouse = 100 m.
• Angle of elevation from ship 1 = 30°.
• Angle of elevation from ship 2 = 45°.
• The ships are on opposite sides of the lighthouse.
• The sea surface is taken to be level.
• Approximate √3 as 1.73 if needed for numerical values.

Concept / Approach:
Let d₁ be the distance from ship 1 to the lighthouse foot and d₂ be the distance from ship 2. For each right-angled triangle, we use:
tan θ = height / distance.We then compute d₁ and d₂ and add them to obtain the separation between the two ships.

Step-by-Step Solution:
Step 1: For ship 1 (30°): tan 30° = 1/√3 = 100 / d₁ ⇒ d₁ = 100√3.Step 2: For ship 2 (45°): tan 45° = 1 = 100 / d₂ ⇒ d₂ = 100 m.Step 3: Distance between the ships = d₁ + d₂ = 100√3 + 100.Step 4: Using √3 ≈ 1.73, compute 100√3 ≈ 173.Step 5: Therefore, separation ≈ 173 + 100 = 273 m.

Verification / Alternative check:
Check with rough sketch: the ship with angle 45° must be closer, so 100 m makes sense. The other ship at distance about 173 m also fits a shallower 30° angle. Summing the two distances gives a realistic total separation.

Why Other Options Are Wrong:
173 m represents only one leg (distance of the farther ship) and ignores the other ship. 200 m or 300 m do not match the sum of 100 and 100√3. 250 m is simply a distractor and does not correspond to any correct trigonometric combination from the given data.

Common Pitfalls:
Common mistakes include subtracting the distances instead of adding them, or assigning the wrong angle to the wrong distance (for example, using 45° for the farther ship). Remember: larger angle → closer distance; and since the ships are on opposite sides, the total separation is the sum of the two distances.

Final Answer:
The distance between the two ships is approximately 273 m.