An observer who is 1.6 m tall is standing at a horizontal distance of 20√3 m from the base of a vertical tower. The angle of elevation from the observer's eye level to the top of the tower is 30°. What is the height of the tower?

Introduction / Context:
This question is a classic example of a height and distance problem involving trigonometry. It uses the concept of angle of elevation and right triangles to determine the height of a tower observed from a certain distance. Such problems commonly appear in aptitude exams, and they test your ability to apply trigonometric ratios, especially tangent, in real life style situations. Accounting for the observer's own height is an important part of accurately computing the total height of the tower.

Given Data / Assumptions:

• Height of the observer (eye level) = 1.6 m.
• Horizontal distance from observer to the base of the tower = 20√3 m.
• Angle of elevation from the observer's eye to the top of the tower = 30°.
• The tower is vertical and the ground is assumed to be horizontal.
• We need to find the total height of the tower from the ground.

Concept / Approach:
In a right triangle formed by the tower, the ground, and the line of sight, the tangent of the angle of elevation is given by: tan(theta) = opposite side / adjacent side. Here, the opposite side is the vertical distance from the observer's eye to the top of the tower, and the adjacent side is the horizontal distance from the observer to the tower. Using the given angle of 30°, and the identity tan(30°) = 1 / √3, we can first find the vertical height above the observer's eye. Then we add the observer's height (1.6 m) to obtain the total height of the tower.

Step-by-Step Solution:
Step 1: Set up the right triangle. Let h be the height of the tower. Height from observer's eye to top of tower = h - 1.6. Horizontal distance from observer to tower = 20√3 m. Step 2: Use the tangent ratio. tan(30°) = (h - 1.6) / (20√3). Step 3: Substitute tan(30°) = 1 / √3. 1 / √3 = (h - 1.6) / (20√3). Step 4: Cross multiply to solve for h - 1.6. h - 1.6 = (20√3) * (1 / √3) = 20. Step 5: Add the observer's height to find total tower height. h = 20 + 1.6 = 21.6 m. Therefore, the height of the tower is 21.6 m.

Verification / Alternative check:
We can verify the result by plugging back into the tangent formula. If h = 21.6 m, then the vertical distance from eye to top is 21.6 - 1.6 = 20 m. The horizontal distance is 20√3 m. So tan(30°) = opposite / adjacent = 20 / (20√3) = 1 / √3, which matches the known value of tan(30°). This confirms that our calculated tower height is consistent with the trigonometric relationship and the given data.

Why Other Options Are Wrong:
23.2 m and 24.72 m: These values produce a vertical difference that does not match tan(30°) when combined with the horizontal distance of 20√3 m, so the trigonometric relation fails.
19.6 m: This height is less than the vertical distance implied by the angle and distance; substituting it back into tan(30°) does not yield 1 / √3.
None of these: This is incorrect because we have found a specific option, 21.6 m, that satisfies all conditions exactly.

Common Pitfalls:
Some students forget to add the observer's height after calculating the vertical distance using trigonometry, thus giving only the height above eye level instead of the full tower height. Others may mix up sine, cosine, and tangent, or misremember that tan(30°) equals 1 / √3. Ensuring a clear diagram, carefully identifying opposite and adjacent sides, and remembering the correct trigonometric values prevents most of these mistakes.

Final Answer:
The height of the tower is 21.6 m.