At a point 129 m from the foot of a cliff on level ground, the angle of elevation of the top of the cliff is 30°. What is the height of the cliff?

Introduction / Context:
This question checks your understanding of basic trigonometry in right angled triangles, specifically the use of the tangent ratio to find the height of a vertical object such as a cliff. You are given the horizontal distance from the observer to the foot of the cliff and the angle of elevation to the top. From this information, you need to determine the height of the cliff using a simple relation involving tan 30 degrees.

Given Data / Assumptions:

• Horizontal distance from the observer to the foot of the cliff = 129 m.
• Angle of elevation to the top of the cliff = 30 degrees.
• The ground between the observer and the foot of the cliff is level.
• The cliff is vertical, so the triangle formed is right angled at the foot.
• Standard trigonometric value: tan 30 degrees = 1 / sqrt(3).

Concept / Approach:
In a right triangle, tan θ = opposite / adjacent. Here, the opposite side is the height of the cliff and the adjacent side is the horizontal distance from the observer to the foot of the cliff (129 m). Rearranging the formula gives height = distance * tan θ. Substituting θ = 30 degrees and tan 30 = 1 / sqrt(3) will provide the exact height in terms of sqrt(3).

Step-by-Step Solution:
Let the height of the cliff be H. We know that tan 30 = H / 129. Substitute tan 30 = 1 / sqrt(3): 1 / sqrt(3) = H / 129. So H = 129 * (1 / sqrt(3)) = 129 / sqrt(3). Rationalise: 129 / sqrt(3) = (129 * sqrt(3)) / 3 = 43 * sqrt(3). Therefore, the height of the cliff is 43√3 m.

Verification / Alternative check:
We know that tan 30 degrees is small (about 0.577). Multiplying 129 by approximately 0.577 gives about 74.5 m. Now 43√3 is roughly 43 * 1.732 which is around 74.5 m as well. So the computed expression 43√3 m is numerically consistent with the approximate calculation and confirms that the result is reasonable.

Why Other Options Are Wrong:

• 50√3 m: This is larger than what tan 30 degrees would give for a distance of 129 m.
• 45√3 m: Slightly too large when compared with the accurate product 129 * tan 30.
• 47√3 m: Also overestimates the height based on the given angle and distance.
• 40√3 m: Underestimates the height and does not satisfy H = 129 * tan 30.

Common Pitfalls:
A common error is to interchange the roles of opposite and adjacent sides, which leads to using H = 129 / tan 30 instead of H = 129 * tan 30. Another frequent mistake is to use approximate values of tan incorrectly or to confuse tan 30 with tan 60. Always remember the exact standard values: tan 30 = 1 / sqrt(3), tan 45 = 1, tan 60 = sqrt(3).

Final Answer:
The height of the cliff is 43√3 m.