As the angle of elevation of the sun decreases from 45° to 30°, the length of the shadow of a vertical pillar increases by 60 m. What is the height of the pillar in metres?

Introduction / Context:
This is a classic heights and distances question involving a vertical pillar and its changing shadow length as the sun moves. As the angle of elevation decreases, the shadow becomes longer. The problem uses two standard angles, 45° and 30°, so exact values of tangents can be used to relate the height of the pillar to the shadow lengths.

Given Data / Assumptions:

• Initial angle of elevation of the sun: 45°.
• Later angle of elevation: 30°.
• Increase in shadow length: 60 m.
• The pillar is vertical and the ground is horizontal and level.

Concept / Approach:
For a vertical pillar of height h and a shadow of length L on horizontal ground, tan θ = h / L, where θ is the angle of elevation of the sun. We set up two equations, one for θ = 45° and one for θ = 30°, with shadow lengths differing by 60 m. Solving this system gives h in terms of √3.

Step-by-Step Solution:
Let h be the height of the pillar. Let L₁ be the shadow length at 45°, and L₂ at 30°. Given that L₂ − L₁ = 60 m. At θ = 45°, tan 45° = h / L₁ ⇒ 1 = h / L₁ ⇒ L₁ = h. At θ = 30°, tan 30° = h / L₂ ⇒ 1/√3 = h / L₂ ⇒ L₂ = √3·h. Now L₂ − L₁ = √3·h − h = h(√3 − 1). Given L₂ − L₁ = 60, so h(√3 − 1) = 60. Therefore h = 60 / (√3 − 1). Rationalise the denominator: h = 60(√3 + 1) / (3 − 1) = 60(√3 + 1)/2. Simplify: h = 30(√3 + 1) metres.
Verification / Alternative check:
You can approximate √3 ≈ 1.732. Then h ≈ 30 × (1.732 + 1) = 30 × 2.732 ≈ 81.96 m. Using this approximate height in the original tangent relations reproduces the 60 m difference in shadow lengths, confirming the solution.

Why Other Options Are Wrong:
Options involving 60(√3 + 1) or 60(√3 − 1) arise from incorrect handling of the equation L₂ − L₁ = 60 or from forgetting to divide by 2 after rationalising. Options with a minus sign in the numerator imply a negative height or a much smaller pillar, which contradicts the geometry and the given increase of 60 m in the shadow.

Common Pitfalls:
Students sometimes swap L₁ and L₂ or use tan θ = L / h instead of h / L. It is also easy to forget to rationalise or to mis-handle √3 − 1 in the denominator. Writing each step clearly and keeping track of which angle corresponds to which shadow length avoids these issues.

Final Answer:
The height of the pillar is 30(√3 + 1) m.