Multiple choice

Observe that, at any instant, the minute and hour hands of a clock make two angles between them whose sum is $360^o$. At $6:15$ what is the difference between these two angles is$?$

  1. $165^o$
  2. $170^o$
  3. $175^o$
  4. $180^o$
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A Correct answer
Explanation

At 6:15, the hour hand is at 6 + 15/60 = 6.25 hours. Position in degrees = 6.25 * 30 = 187.5 degrees. The minute hand is at 15 minutes, which is 15 * 6 = 90 degrees. The difference is 187.5 - 90 = 97.5 degrees. However, the question asks for the difference between the two angles (the reflex and the acute), which is 360 - 2*97.5 = 165 degrees.

AI explanation

Using the clock angle formula, the angles formed by the hands at 6:15 are calculated as 30 degrees per hour minus 5.5 degrees per minute. Substituting the values gives (30 * 6) - (5.5 * 15) = 180 - 82.5 = 97.5 degrees, making the other angle 262.5 degrees. The difference between these two angles is 262.5 - 97.5 = 165 degrees.

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