Two ships are sailing in the sea on opposite sides of a vertical lighthouse. From the decks of the two ships, the angles of elevation of the top of the lighthouse are observed to be 30° and 45° respectively. If the lighthouse is 100 m high (take √3 = 1.73 where needed), find the distance between the two ships.

Introduction / Context:
This question involves two ships located on opposite sides of a vertical lighthouse at sea. Observers on each ship measure the angle of elevation of the top of the lighthouse. One ship observes an angle of 30°, while the other observes an angle of 45°. Since the height of the lighthouse is known, we can treat each observation as a separate right angled triangle sharing the same vertical height but having different horizontal distances from the lighthouse. By using basic trigonometric ratios, we can find the distance of each ship from the lighthouse and then add these distances to obtain the distance between the two ships.

Given Data / Assumptions:

- The lighthouse is vertical and 100 m high. - Two ships are on opposite sides of the lighthouse. - Angle of elevation from the first ship is 30°. - Angle of elevation from the second ship is 45°. - Sea level is considered to be a horizontal plane. - We use tan 30° = 1 / √3 and tan 45° = 1.

Concept / Approach:
The height of the lighthouse is the same for both ships, so we form two right angled triangles that share the same vertical side (100 m) but have different horizontal bases. In each triangle, the tangent of the angle of elevation is equal to the height divided by the horizontal distance from the lighthouse. For the ship with a 45° angle, the calculation is particularly simple because tan 45° = 1, so its distance from the lighthouse equals the height. For the ship with a 30° angle, we use tan 30° = 1 / √3 to find a larger horizontal distance. The total distance between the ships is the sum of these two horizontal distances because they lie on opposite sides of the lighthouse.

Step-by-Step Solution:
Step 1: Let d1 be the distance of the ship where the angle of elevation is 30°. Step 2: Let d2 be the distance of the ship where the angle of elevation is 45°. Step 3: For the 45° angle, tan 45° = 100 / d2. Since tan 45° = 1, we get 1 = 100 / d2, so d2 = 100 m. Step 4: For the 30° angle, tan 30° = 100 / d1. Using tan 30° = 1 / √3, we have 1 / √3 = 100 / d1, so d1 = 100√3 m. Step 5: The distance between the two ships is d1 + d2 = 100√3 + 100. Step 6: If we use √3 ≈ 1.73, then 100√3 ≈ 173 m, so d1 + d2 ≈ 173 + 100 = 273 m. Step 7: Therefore, the distance between the two ships is 273 m.

Verification / Alternative check:
We can verify the result numerically. If one ship is 100 m from the lighthouse, then the angle of elevation is tan⁻¹(100 / 100) = tan⁻¹(1) = 45°, which is correct. If the other ship is at 100√3 m, then the angle of elevation is tan⁻¹(100 / (100√3)) = tan⁻¹(1 / √3) = 30°, which also matches the given data. Since both angles are satisfied with these distances and the sum of the distances matches one of the answer options, the result is consistent.

Why Other Options Are Wrong:
Option A (173 m) corresponds roughly to 100√3 alone and ignores the distance of the second ship. Option B (200 m) underestimates the total distance, as the ships are clearly more than 100 m apart. Option D (300 m) is larger than the computed value and does not follow from tan 30° and tan 45° calculations. Only 273 m correctly represents d1 + d2 when both tangent relations are applied correctly.

Common Pitfalls:
Students sometimes forget that the ships are on opposite sides of the lighthouse and mistakenly subtract the distances instead of adding them. Another common mistake is using sine or cosine instead of tangent when the height and horizontal distance are involved. Mixing up which angle corresponds to which distance or using an incorrect value for √3 can also lead to errors. Being careful with which sides of the right triangle correspond to opposite and adjacent and remembering that distance between the ships is the sum of their distances from the lighthouse avoids these issues.

Final Answer:
The distance between the two ships is 273 m, which corresponds to option C.