A spinning disc, Disc A, rotates at a constant speed of 3pi radians per second. A second disc, Disc B,...

1. TRANSLATE the problem information

• Given information:
  • Disc A rotates at \(3\pi\) radians per second
  • Disc B rotates \(\frac{5\pi}{2}\) radians per second faster than Disc A
  • Need complete revolutions in one minute
  • 1 revolution = \(2\pi\) radians

• What this tells us: We need to find Disc B's speed first, then convert units

2. INFER the solution approach

• We need two main steps: find Disc B's speed, then convert units
• For unit conversion, we'll need to change both time units (seconds to minutes) and angular units (radians to revolutions)

3. SIMPLIFY to find Disc B's rotational speed

• Disc B's speed = Disc A's speed + \(\frac{5\pi}{2}\)
• Disc B's speed = \(3\pi + \frac{5\pi}{2}\)

• To add these, convert to common denominator:
  • \(3\pi = \frac{6\pi}{2}\)
  • So: \(\frac{6\pi}{2} + \frac{5\pi}{2} = \frac{11\pi}{2}\) radians per second

4. INFER the unit conversion strategy

• We need to go from radians/second to revolutions/minute
• This requires converting seconds to minutes (multiply by 60) and radians to revolutions (divide by \(2\pi\))

5. SIMPLIFY the unit conversion

• Speed in rev/min = \(\frac{11\pi}{2}\) rad/s × 60 s/min × \(\frac{1\text{ rev}}{2\pi\text{ rad}}\)
• \(= \frac{11\pi \times 60}{2 \times 2\pi}\) rev/min
• \(= \frac{660\pi}{4\pi}\) rev/min
• The \(\pi\) terms cancel: \(= \frac{660}{4} = 165\) rev/min (use calculator)

Answer: C (165)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students misinterpret "\(\frac{5\pi}{2}\) radians per second faster" and might subtract instead of add, or add incorrectly.

They might calculate Disc B's speed as \(3\pi - \frac{5\pi}{2} = \frac{6\pi}{2} - \frac{5\pi}{2} = \frac{\pi}{2}\) rad/s, leading to a much smaller final answer. Converting this would give 15 revolutions per minute, which doesn't match any answer choice, causing confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students get the right speed for Disc B (\(\frac{11\pi}{2}\) rad/s) but make errors in the unit conversion process.

A common mistake is forgetting one of the conversion factors - either forgetting to multiply by 60 for the time conversion or forgetting to divide by \(2\pi\) for the angular conversion. This leads to answers that are off by factors of 60 or \(2\pi\), potentially leading them to select Choice A (90) or other incorrect options.

The Bottom Line:

This problem tests both careful language interpretation and systematic unit conversion. Students need to methodically apply both time and angular unit conversions while maintaining algebraic precision throughout.