A vertical tower has a height of 300 metres. When its top is viewed from the top of another smaller vertical tower, the angle of depression is 60 degrees. The horizontal distance between the bases of the two towers is 120 metres. What is the height of the smaller tower in metres?

Introduction / Context:
Here we have two vertical towers standing on the same horizontal ground. The taller tower has known height, and from its top an observer sees the top of the smaller tower at a given angle of depression. The horizontal distance between the two towers is also given. Using the connection between angle of depression and angle of elevation, and basic trigonometry, we can calculate the height of the smaller tower.

Given Data / Assumptions:

• The height of the taller tower is 300 metres.
• The angle of depression from the top of the taller tower to the top of the smaller tower is 60 degrees.
• The horizontal distance between the bases of the two towers is 120 metres.
• Both towers stand vertically on the same level ground.
• We ignore effects such as earth curvature and treat the situation as a right angled triangle between tower tops and bases.

Concept / Approach:
An angle of depression from a horizontal line of sight at the top of one tower equals the angle of elevation from the corresponding point on the ground or from the other top. Thus, the angle of elevation of the top of the taller tower from the top of the smaller tower is also 60 degrees. Let the height of the smaller tower be h metres. The vertical difference in height between the two tower tops is 300 − h, and the horizontal distance between them is 120 metres. Using tan 60 degrees equals vertical difference divided by horizontal distance, we can solve for h.

Step-by-Step Solution:
Step 1: Let h be the height of the smaller tower in metres. Step 2: The difference in vertical height between the two tops is 300 − h metres. Step 3: The horizontal distance between the towers is given as 120 metres. Step 4: From the top of the taller tower, angle of depression to the top of the smaller tower is 60°. Thus, tan 60° = (300 − h) / 120. Step 5: Since tan 60° = √3, we have √3 = (300 − h) / 120, so 300 − h = 120√3. Step 6: Rearrange to find h = 300 − 120√3. Using √3 ≈ 1.732, h ≈ 300 − 120 × 1.732 ≈ 300 − 207.84 = 92.16 metres, which is approximately 92.15 metres.

Verification / Alternative check:
Check back in the tangent relation. Vertical difference 300 − 92.16 ≈ 207.84 metres. Then tan 60° should be 207.84 / 120 ≈ 1.732, which matches √3. The geometry is consistent with a right triangle having base 120 and opposite side approximately 207.84 for a 60 degree angle, confirming the solution.

Why Other Options Are Wrong:
Options 88.24, 106.71, and 112.64 correspond to different vertical differences that do not lead to tan 60° when divided by 120. Option 100 would give a difference of 200 metres, making tan 60° approximately 1.666, which is not equal to √3. Only about 92.15 metres satisfies all conditions of the problem.

Common Pitfalls:
One common mistake is to misinterpret angle of depression and to use 300 + h as the vertical distance instead of 300 − h. Another error is to confuse distance between bases with distance between tops. Some learners also make algebraic mistakes when isolating h, or approximate √3 too roughly, which can mislead them towards an incorrect option when numerical values are close.

Final Answer:
The height of the smaller tower is approximately 92.15 metres.