Difficulty: Easy
Correct Answer: 43√3 metre
Explanation:
Introduction:
This is a straightforward height and distance question. A cliff is observed from a known horizontal distance, and the angle of elevation of its top is given. Using basic trigonometry, we can compute the height of the cliff from these data.
Given Data / Assumptions:
Concept / Approach:
In a right-angled triangle formed by the cliff height, the ground distance, and the line of sight, the tangent of the angle of elevation is:
tan θ = (height of cliff) / (horizontal distance).Here, θ = 30°, horizontal distance = 129 m. We use tan 30° = 1/√3 to find the height h.
Step-by-Step Solution:
Step 1: Let h be the cliff height.Step 2: Use tan 30° = h / 129.Step 3: Substitute tan 30° = 1/√3, so 1/√3 = h / 129.Step 4: Solve for h: h = 129 / √3.Step 5: Rationalise the denominator: h = (129√3) / 3 = 43√3 m.
Verification / Alternative check:
Approximate numerically with √3 ≈ 1.732: h ≈ 43 * 1.732 ≈ 74.5 m. Check tan 30°: h / 129 ≈ 74.5 / 129 ≈ 0.577 / 1.732 ≈ 1/√3, which confirms that the angle is indeed 30° and that the calculation is consistent.
Why Other Options Are Wrong:
45√3 m, 47√3 m, or 50√3 m correspond to different heights that would yield tangents larger than 1/√3 when divided by 129 m. 43 m is too small and does not fit tan 30°. Only 43√3 m satisfies the relationship h = 129 * tan 30° exactly.
Common Pitfalls:
Students sometimes use sine or cosine instead of tangent for height-and-distance problems, or they forget to rationalise the denominator, leaving the answer as 129/√3. While 129/√3 is correct, the simplified form 43√3 is usually preferred and also matches the options given in such exams.
Final Answer:
The height of the cliff is 43√3 metres.
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