A wheel rotates counterclockwise by 7pi/12 radians and then rotates an additional 110 degrees in the same direction. What is...

1. TRANSLATE the problem information

• Given information:
  • First rotation: \(\frac{7\pi}{12}\) radians counterclockwise
  • Second rotation: \(110\) degrees counterclockwise
  • Need: Total rotation in degrees

• What this tells us: We need to convert radians to degrees, then add the angles.

2. INFER the approach

• Since we need the answer in degrees, convert the radian measurement first
• Both rotations are counterclockwise (same direction), so we'll add them together
• Use the radian-to-degree conversion formula

3. SIMPLIFY the radian conversion

• Apply conversion formula: \(\mathrm{radians} \times \frac{180°}{\pi} = \mathrm{degrees}\)
• \(\frac{7\pi}{12} \times \frac{180°}{\pi} = ?\)
• The \(\pi\) cancels: \(\frac{7}{12} \times 180° = \frac{7 \times 180°}{12} = 7 \times 15° = 105°\)

4. INFER the final calculation

• Both rotations are in same direction → add the angles
• Total rotation = \(105° + 110° = 215°\)

Answer: D. 215

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misread "same direction" and subtract instead of add the rotations.

They think: "The wheel rotated \(\frac{7\pi}{12}\) radians, then went back 110 degrees" and calculate \(105° - 110° = -5°\). Since negative doesn't make sense, they might take the absolute value and get 5°, leading to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make arithmetic errors in the radian conversion.

Common mistake: \(\frac{7\pi}{12} \times \frac{180°}{\pi} = \frac{7 \times 180°}{12} = 7 \times 12° = 84°\) instead of \(7 \times 15° = 105°\). Then they calculate \(84° + 110° = 194°\), which isn't an answer choice. This leads to confusion and guessing.

The Bottom Line:

This problem tests both unit conversion skills and careful reading comprehension. Students must correctly execute the radian-to-degree conversion AND recognize that "same direction" means the rotations add together.