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.
Aptitude
Height and Distance
Difficulty: Medium
Choose an option
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A20 √3 m
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B25/ √3 m
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C25 √3 m
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D20 √3 m
Answer
Correct Answer: 25 √3 m
Explanation
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:
s30 − s60 = H(√3 − 1/√3) = 50.√3 − 1/√3 = (3 − 1)/√3 = 2/√3.Thus H * (2/√3) = 50 → H = 25 √3 m.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