Clock – Right angles between 5 and 6: At what times between 5:00 and 6:00 will the hour and minute hands be at right angles (90°)?
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A10 10/11 min and 43 7/11 min past 5
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B9 7/11 min and 50 3/11 min past 5
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C11 1/11 min past 5 and 10 10/11 min past 6
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D43 7/11 min past 5 and 10 10/11 min past 6
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ENone of these
Answer
Correct Answer: 10 10/11 min and 43 7/11 min past 5
Explanation
Introduction / Context:Right-angle (90°) problems require setting the angle between the hour and minute hands to 90°. Between 5:00 and 6:00 there are generally two instants where the condition holds.
Given Data / Assumptions:
- Time window: 5:00 to 6:00.
- Minute-hand speed: 6° per minute.
- Hour-hand speed: 0.5° per minute; position at 5:00 is 150°.
Concept / Approach:Angle formula: |30h − 5.5m| where h is the hour and m is minutes past the hour. For h=5, solve |150 − 5.5m| = 90.
Step-by-Step Solution:1) Case A: 150 − 5.5m = 90 → 5.5m = 60 → m = 10 10/11 minutes.2) Case B: 150 − 5.5m = −90 → 5.5m = 240 → m = 43 7/11 minutes.3) Both moments lie within (5:00, 6:00).
Verification / Alternative check:Plug m back into angle formula to confirm 90° (within fractional precision).
Why Other Options Are Wrong:Options involving times past 6:00 or mis-stated fractions do not satisfy the 5:00–6:00 window or the 90° condition.
Common Pitfalls:Using 6°−0.5° = 5.5° difference correctly but mixing signs; forgetting there are two solutions.
Final Answer:10 10/11 min and 43 7/11 min past 5