You are given below a question followed by two statements I and II. You have…
2023
You are given below a question followed by two statements I and II. You have to decide whether the data provided in the statement(s) is/are sufficient to answer the question.
Question: A man standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is 60°. When he moves away from the bank, he finds the change in angle. Find the breadth of the river.
Statements:
I. The changed angle is found to be 30°.
II. He noticed that when he moved 40 m away from the bank, the tree subtended a 30° angle on the other side of the river.
- A.
If data in statement I alone is sufficient to answer the question
- B.
If data in statement II alone is sufficient to answer the question
- C.
If data in both statements I & II together are necessary to answer the question
- D.
If data in both statements I & II together are not sufficient to answer the question
- E.
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Show answer & explanation
Correct answer: B
To determine if the provided statements are sufficient to find the breadth of the river, we can model this as a trigonometry problem. Let the breadth of the river be B, and the height of the tree be H.
Step-by-Step Analysis
Understand the initial state: The man observes a tree on the opposite bank at an angle of 60 degrees. Using trigonometry, the relationship between the height of the tree, the breadth of the river, and the angle is:
tan(60 degrees) = H / B
This gives us H = B * tan(60 degrees) = B * sqrt(3).
Evaluate Statement I: "The changed angle is found to be 30 degrees."
This tells us the new angle but does not provide the distance moved away from the bank. Without the distance, we have two unknowns (Breadth B and Height H) and cannot solve the equation. This statement alone is insufficient.
Evaluate Statement II: "He noticed that when he moved 40 m away from the bank, the tree subtended a 30 degree angle on the other side of the river."
This statement provides the distance moved (40 m) and the new angle (30 degrees).
New distance from the bank = B + 40
New angle relationship: tan(30 degrees) = H / (B + 40)
Since H = B * sqrt(3), we can substitute:
tan(30 degrees) = (B * sqrt(3)) / (B + 40)
1/sqrt(3) = (B * sqrt(3)) / (B + 40)
B + 40 = 3 * B
40 = 2 * B
B = 20 m.
Since Statement II provides both the distance moved and confirms the change in angle, it contains all the information needed to solve for B. Statement I is redundant because Statement II already includes the necessary information.