Shadow difference for sun altitudes 30° and 60°: A vertical tower on level ground casts shadows when the sun’s altitude is 30° and 60°. The longer shadow is 50 m more than the shorter one. Find the height of the tower.

Introduction / Context:
Shadow length s = H cot θ for a vertical object. Two altitudes give two shadows; their difference is known, allowing H to be solved exactly.

Given Data / Assumptions:

• Altitudes: 30° and 60°.
• Difference in shadow lengths = 50 m.
• Tower height = H.

Concept / Approach:
s30 = H cot 30° = H √3; s60 = H cot 60° = H / √3. Their difference equals 50 → solve for H.

Step-by-Step Solution:

Verification / Alternative check:
Compute shadows with H = 25√3: s30 = 75; s60 = 25 → difference 50 ✔️.

Why Other Options Are Wrong:
20√3 and 25/√3 give wrong differences; duplicate 20√3 is a distractor.

Common Pitfalls:
Using tan instead of cot for shadow; subtracting in reverse (negative); arithmetic slip when simplifying 2/√3.

Final Answer:
25 √3 m