The tops of two vertical poles of heights 60 metres and 35 metres are connected by a straight rope. The rope makes an angle with the horizontal such that tan of this angle is 5/9. What is the horizontal distance between the two poles in metres?

Introduction / Context:
This question involves two vertical poles of different heights connected at their tops by a rope that is inclined to the horizontal. The tangent of the angle that the rope makes with the horizontal is given. The difference in heights of the poles and the tangent value allow us to compute the horizontal distance between the poles using basic trigonometry and the definition of tangent.

Given Data / Assumptions:

• One pole has a height of 60 metres.
• The other pole has a height of 35 metres.
• The tops of the poles are joined by a straight rope.
• The rope makes an angle θ with the horizontal such that tan θ = 5 / 9.
• The poles stand vertically on the same level ground.

Concept / Approach:
Let the horizontal distance between the two poles be d metres. The vertical difference between their tops is equal to the difference in their heights. The rope is the slant side of a right angled triangle whose opposite side is this vertical difference and whose adjacent side is the horizontal distance d. Since tan θ is defined as opposite side divided by adjacent side, we can write tan θ = (vertical difference) / d. Substituting the known values, we can solve for d.

Step-by-Step Solution:
Step 1: Calculate the vertical difference in height between the two poles: 60 − 35 = 25 metres. Step 2: Let d be the horizontal distance between the two poles in metres. Step 3: From the given information, tan θ = 5 / 9 and tan θ also equals vertical difference divided by horizontal distance, so 5 / 9 = 25 / d. Step 4: Cross multiply to obtain 5d = 9 × 25 = 225. Step 5: Solve for d: d = 225 / 5 = 45 metres.

Verification / Alternative check:
Check the tangent value using the found distance. With opposite side 25 metres and adjacent side 45 metres, tan θ = 25 / 45 = 5 / 9, which exactly matches the given condition. This confirms that the horizontal distance has been computed correctly. The geometry matches a right angled triangle formed by the difference in heights and the distance between the poles.

Why Other Options Are Wrong:
If the distance were 63 metres or 30 metres, the tangent of the angle formed by the rope would be 25 / 63 or 25 / 30, neither of which equals 5 / 9. A distance of 25 metres would lead to tan θ = 1, and a distance of 40 metres would give tan θ = 25 / 40 = 5 / 8. None of these satisfy the given condition tan θ = 5 / 9.

Common Pitfalls:
A common error is to use the sum of the heights instead of the difference, but the rope connects the tops of the poles, so only their height difference forms the opposite side. Another mistake is to invert the ratio and use d / 25 rather than 25 / d for tan θ. Students may also confuse which segment is horizontal and which is vertical in the right angled triangle. Drawing a clear diagram often prevents these misunderstandings.

Final Answer:
The horizontal distance between the two poles is 45 metres.