From the top of a vertical tower 60 m high, the angles of depression of the top and the bottom of a vertical pole are observed to be 45° and 60° respectively. If the pole and the tower stand on the same horizontal plane, what is the height of the pole (in metres)?

Introduction / Context:
This height and distance question involves angles of depression from the top of a tower to both the top and the bottom of a shorter pole standing some distance away on the same level ground. Angles of depression from a horizontal line of sight are equal to angles of elevation from the point on the ground to the observer. By converting these angles of depression into angles of elevation and using tangent ratios, we can relate the heights and the horizontal distance between the tower and the pole. This allows us to determine the unknown height of the pole using simple trigonometry.

Given Data / Assumptions:

- The tower is vertical and 60 m high. - A vertical pole stands on the same horizontal plane as the tower. - From the top of the tower, the angle of depression to the top of the pole is 45°. - From the top of the tower, the angle of depression to the bottom (foot) of the pole is 60°. - The horizontal distance between the tower and the pole is unknown. - Ground is level, and both structures are perpendicular to the ground.

Concept / Approach:
Let h be the height of the pole and d be the horizontal distance between the tower and the pole. The line of sight from the top of the tower to the top of the pole forms a right angled triangle whose vertical side is the difference in heights (60 − h) and whose base is d. The angle at the tower top with respect to the horizontal is 45°. Similarly, the line of sight from the top of the tower to the bottom of the pole forms another right angled triangle with vertical side 60 and base d and an angle of 60°. Using tan 45° and tan 60° on these two triangles gives us two expressions involving d and h. Solving these simultaneously allows us to find the exact height of the pole.

Step-by-Step Solution:
Step 1: Let h be the height of the pole in metres, and let d be the horizontal distance between the tower and the pole. Step 2: Consider the angle of depression of 60° from the top of the tower to the bottom of the pole. The vertical opposite side is the full height of the tower, 60 m, and the horizontal adjacent side is d. Step 3: Using the tangent ratio, tan 60° = 60 / d. Since tan 60° = √3, we get √3 = 60 / d, so d = 60 / √3. Step 4: Simplify d: d = 60 / √3 = (60√3) / 3 = 20√3. Step 5: Now consider the angle of depression of 45° from the top of the tower to the top of the pole. The vertical difference in height between the top of the tower and the top of the pole is 60 − h, and the horizontal distance is still d. Step 6: Using tan 45° = (60 − h) / d and tan 45° = 1, we have 1 = (60 − h) / d, so 60 − h = d. Step 7: Substitute d = 20√3 into 60 − h = d to obtain 60 − h = 20√3. Step 8: Rearranging gives h = 60 − 20√3. Step 9: Factor out 20 from the expression: h = 20(3 − √3). Step 10: Therefore, the exact height of the pole is 20(3 − √3) m.

Verification / Alternative check:
We can check this result numerically. Using √3 ≈ 1.732, h ≈ 20(3 − 1.732) = 20 * 1.268 ≈ 25.36 m. Then the difference in heights is 60 − h ≈ 34.64 m and the distance d is 20√3 ≈ 34.64 m. For the line of sight to the top of the pole, tan of the angle is (60 − h) / d ≈ 34.64 / 34.64 = 1, which corresponds to 45°. For the line of sight to the bottom of the pole, tan of the angle is 60 / d ≈ 60 / 34.64 ≈ 1.732, which corresponds to √3 and hence to 60°. Both angles match the given information, confirming that the height 20(3 − √3) m is correct.

Why Other Options Are Wrong:
Option B (20(3 + √3) m) is larger than 60 m when evaluated, which is impossible for the pole height since the tower height is only 60 m. Option C (20(√3 + 1) m) gives a height that, when substituted into 60 − h, does not equal the computed horizontal distance 20√3, so it fails the tan 45° relation. Option D (20(3√3) m) is much larger than 60 m and is clearly inconsistent with the given tower height. Only 20(3 − √3) m satisfies both tangent equations exactly and is less than the tower height, as expected for a shorter pole.

Common Pitfalls:
A typical mistake is to misinterpret angles of depression as angles of elevation without correctly reflecting them across the horizontal line. This can lead to swapping the roles of vertical and horizontal distances or incorrectly identifying which angle corresponds to which segment. Another error is forgetting that both lines of sight use the same horizontal distance d, causing students to introduce multiple distances unnecessarily. Algebraic errors can also occur when simplifying d = 60 / √3 or when solving for h in 60 − h = d. Careful diagram drawing and step-by-step use of tan θ = opposite / adjacent for each angle help avoid these problems.

Final Answer:
The height of the pole is 20(3 − √3) m, which corresponds to option A.