A vertical pole is 15 m high. If the Sun’s elevation (altitude) changes from 30° to 60°, what is the difference (in metres) between the lengths of the shadows of the pole corresponding to these two angles of elevation?

Introduction:
This question tests your understanding of how the length of a shadow of a vertical object changes with the Sun’s angle of elevation. Using basic trigonometry, we express the shadow length in terms of the height and the tangent of the Sun’s altitude for each case, and then find the difference between the two shadow lengths.

Given Data / Assumptions:

• Height of the vertical pole = 15 m.
• First altitude of the Sun = 30°.
• Second altitude of the Sun = 60°.
• The pole stands perfectly vertical on level ground.

Concept / Approach:
For a vertical object of height h and Sun’s altitude θ, the shadow length L on level ground is given by:
tan θ = h / L ⇒ L = h / tan θ.We compute L₁ for θ = 30° and L₂ for θ = 60°, then find the difference L₁ − L₂.

Step-by-Step Solution:
Step 1: Let h = 15 m.Step 2: For θ = 30°, tan 30° = 1/√3.Step 3: Shadow length L₁ = h / tan 30° = 15 / (1/√3) = 15√3.Step 4: For θ = 60°, tan 60° = √3.Step 5: Shadow length L₂ = h / tan 60° = 15 / √3 = (15√3)/3 = 5√3.Step 6: Difference in shadow lengths: L₁ − L₂ = 15√3 − 5√3 = 10√3 m.

Verification / Alternative check:
Using √3 ≈ 1.732, approximate L₁ ≈ 15 * 1.732 ≈ 25.98 m, L₂ ≈ 5 * 1.732 ≈ 8.66 m. The difference is about 17.32 m, which equals 10 * 1.732 ≈ 10√3 m, matching our exact expression.

Why Other Options Are Wrong:
7.5 m and 15 m ignore the trigonometric dependence on the angles and are far smaller than the true difference. 5√3 m is only the shorter shadow length itself, not the difference. 20 m does not correspond to any correct trigonometric combination derived from the given angles and pole height.

Common Pitfalls:
Students sometimes incorrectly subtract the heights instead of the shadow lengths or confuse tan θ with its reciprocal. Remember: for a fixed height, as the angle of elevation increases, the shadow becomes shorter. Correctly using L = h / tan θ for each angle avoids such errors.

Final Answer:
The difference between the lengths of the shadows is 10√3 m.