Question

A ship was moving towards lighthouse on the shore with uniform speed. A man standing on the lighthouse observes the ship. Eight minutes later, he finds that the angle of depression was 45°. In how much time does the ship reach the lighthouse?
Statement 1: Initially, the man standing on the lighthouse observes the ship at an angle of depression of 30°
Statement 2: The speed of the ship was 6 km/h and the height of the lighthouse is 300 m
Directions: This question has a problem and two statements numbered (1) and (2) giving certain information. You have to decide if the information given in the statements is sufficient for answering the problem. Indicate your answer

• A. statement (1) alone is sufficient to answer the question
• B. statement (2) alone is sufficient to answer the question
• C. both the statements together are needed to answer the question
• D. either statement (1) alone or statement (2) alone is sufficient to answer the question
• E. neither statement (1) nor statement (2) suffices to answer the question

Answer 1

The Correct Option is D

To solve the problem, we need to determine if either statement provides sufficient information to find the time the ship takes to reach the lighthouse.

Let's consider the given statements individually:

1. Statement 1: Initially, the man standing on the lighthouse observes the ship at an angle of depression of 30°.
   • From this statement, we know the initial angle of depression, which allows us to find the initial distance using the trigonometric identity:

\[\tan(30^\circ) = \frac{{\text{height of lighthouse}}}{\text{initial distance}}\]

• Since the angle of depression changes to 45° after 8 minutes, we can use the new angle to find the distance at that point:
• Because we have an initial and a subsequent distance, we can find the time to reach the lighthouse using relative speed calculation. Therefore, statement 1 is sufficient.

1. Statement 2: The speed of the ship was 6 km/h and the height of the lighthouse is 300 m.
   • Given the speed of the ship and the height of the lighthouse, when the angle of depression is 45°, the horizontal distance equals the height of the lighthouse (same reasoning as with statement 1).
   • Therefore, the time to cover this final distance at constant speed (6 km/h) can be calculated directly. This makes statement 2 sufficient by itself.

In conclusion, either statement (1) alone or statement (2) alone is sufficient to answer the question. Thus, the correct answer is:

either statement (1) alone or statement (2) alone is sufficient to answer the question