A 1.6 m tall observer stands 45 m away from the foot of a vertical tower. If 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 in metres?

Introduction / Context: This is a standard height and distance problem in trigonometry. It involves using the tangent of an angle of elevation to relate the vertical height difference and the horizontal distance. The observer's eye level must be included to obtain the total height of the tower, a detail that is often overlooked by students. Given Data / Assumptions:

• Height of observer = 1.6 m.
• Horizontal distance between observer and tower = 45 m.
• Angle of elevation from eye level to top of tower = 30°.
• The tower is vertical and the ground is level.
• We use tan 30° = 1/√3 and approximate √3 ≈ 1.732 when needed.

Concept / Approach: Key ideas:

• In a right triangle, tan θ = opposite side / adjacent side.
• The opposite side here is the vertical height difference between the top of the tower and the observer's eye level.
• The adjacent side is the horizontal distance from the observer to the base of the tower, which is 45 m.
• Add the observer's own height to this vertical difference to get the full height of the tower.

Step-by-Step Solution: Step 1: Let H be the total height of the tower in metres. Step 2: The vertical height difference between the top of the tower and the observer's eye level is H − 1.6. Step 3: Using tan 30° = opposite / adjacent = (H − 1.6) / 45. Step 4: Since tan 30° = 1/√3, we have (H − 1.6) / 45 = 1 / √3. Step 5: Rearrange to get H − 1.6 = 45 / √3. Step 6: Simplify 45 / √3 by rationalising if desired: 45 / √3 = (45√3) / 3 = 15√3. Step 7: Approximate √3 ≈ 1.732, so H − 1.6 ≈ 15 × 1.732 = 25.98. Step 8: Therefore H ≈ 25.98 + 1.6 = 27.58 metres. Verification / Alternative check: Step 1: Take H = 27.58 m and recompute the angle of elevation using tan θ = (H − 1.6) / 45. Step 2: Then H − 1.6 ≈ 25.98, so tan θ ≈ 25.98 / 45 ≈ 0.5773. Step 3: tan 30° is exactly 1/√3 ≈ 0.5773, so the computed value is consistent with a 30° angle. Why Other Options Are Wrong: Option 25.98 corresponds only to the vertical rise above the observer's eye and ignores the 1.6 m eye level height. Option 26.58 is too small and does not match the precise calculation using tan 30°. Option 27.98 is slightly larger than the correct sum and would give a tangent less than 1/√3. Option 29.58 is much larger and clearly inconsistent with the trigonometric relationship. Common Pitfalls: A very common mistake is forgetting to add the observer's height to the vertical difference, leading to the answer 25.98 instead of the correct total height. Some students use sin or cos instead of tan, which does not fit this right triangle configuration. Rounding errors or using an incorrect approximate value for √3 can slightly shift the final decimal, so careful approximation is important. Final Answer: The height of the tower is approximately 27.58 metres.