Question

after hillary realized her clock was running 30 minutes fast, she used a knob on the clock to turn the minute hand back 30 minutes. which describes the rotation of the minute hand?

\\(180^\circ\\) clockwise rotation
\\(90^\circ\\) clockwise rotation
\\(90^\circ\\) counterclockwise rotation
\\(180^\circ\\) counterclockwise rotation

Explanation:

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<reasoning>

Determine the direction of rotation

To turn a clock hand back (earlier in time), we must rotate it in the direction opposite to its normal movement. Since a clock normally runs clockwise, turning the hand back requires a counterclockwise rotation.

Calculate the angle of rotation

A full circle represents \(360^\circ\) and corresponds to \(60\) minutes on a clock face.
\[
\text{Angle per minute} = \frac{360^\circ}{60} = 6^\circ/\text{minute}
\]
For a change of \(30\) minutes:
\[
\text{Angle} = 30 \times 6^\circ = 180^\circ
\]

Combine direction and angle

Combining the counterclockwise direction with the calculated angle of \(180^\circ\) yields a \(180^\circ\) counterclockwise rotation.
</reasoning>

<answer>
<mcq-option>(A) \(180^\circ\) clockwise rotation</mcq-option>
<mcq-option>(B) \(90^\circ\) clockwise rotation</mcq-option>
<mcq-option>(C) \(90^\circ\) counterclockwise rotation</mcq-option>
<mcq-correct>(D) \(180^\circ\) counterclockwise rotation</mcq-correct>
</answer>

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Answer:

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<reasoning>

Determine the direction of rotation

To turn a clock hand back (earlier in time), we must rotate it in the direction opposite to its normal movement. Since a clock normally runs clockwise, turning the hand back requires a counterclockwise rotation.

Calculate the angle of rotation

A full circle represents \(360^\circ\) and corresponds to \(60\) minutes on a clock face.
\[
\text{Angle per minute} = \frac{360^\circ}{60} = 6^\circ/\text{minute}
\]
For a change of \(30\) minutes:
\[
\text{Angle} = 30 \times 6^\circ = 180^\circ
\]

Combine direction and angle

Combining the counterclockwise direction with the calculated angle of \(180^\circ\) yields a \(180^\circ\) counterclockwise rotation.
</reasoning>

<answer>
<mcq-option>(A) \(180^\circ\) clockwise rotation</mcq-option>
<mcq-option>(B) \(90^\circ\) clockwise rotation</mcq-option>
<mcq-option>(C) \(90^\circ\) counterclockwise rotation</mcq-option>
<mcq-correct>(D) \(180^\circ\) counterclockwise rotation</mcq-correct>
</answer>

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