Grid-in Question:In the xy-plane, a circle with its center at the origin has a radius of 6. The area of...

1. TRANSLATE the problem information

• Given information:
  • Circle has radius \(\mathrm{r = 6}\)
  • Central angle is \(\mathrm{150}\) degrees
  • Need to find \(\mathrm{k}\) where \(\mathrm{area = k\pi}\)
• What this tells us: We need to find the area of a sector using the given radius and central angle.

2. INFER the approach needed

• To find sector area, we need the formula \(\mathrm{A = \frac{1}{2}r^2\theta}\)
• Critical insight: This formula requires \(\mathrm{\theta}\) to be in radians, not degrees
• Strategy: First convert the angle to radians, then apply the formula

3. TRANSLATE degrees to radians

• Use conversion formula: \(\mathrm{radians = degrees \times \frac{\pi}{180}}\)
• \(\mathrm{150° \times \frac{\pi}{180} = \frac{150\pi}{180}}\)
• SIMPLIFY: \(\mathrm{\frac{150}{180} = \frac{5}{6}}\), so we get \(\mathrm{\frac{5\pi}{6}}\) radians

4. SIMPLIFY using the sector area formula

• \(\mathrm{A = \frac{1}{2}r^2\theta}\)
• \(\mathrm{A = \frac{1}{2}(6)^2(\frac{5\pi}{6})}\)
• \(\mathrm{A = \frac{1}{2}(36)(\frac{5\pi}{6})}\)
• \(\mathrm{A = 18 \times \frac{5\pi}{6}}\)
• \(\mathrm{A = \frac{18 \times 5\pi}{6} = \frac{90\pi}{6} = 15\pi}\)

5. INFER the final answer

• Since the area is \(\mathrm{15\pi}\) and we need it in the form \(\mathrm{k\pi}\)
• Comparing \(\mathrm{k\pi = 15\pi}\), we get \(\mathrm{k = 15}\)

Answer: 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students use degrees directly in the sector area formula without converting to radians.

They calculate \(\mathrm{A = \frac{1}{2}(6)^2(150)}\)

\(\mathrm{= \frac{1}{2}(36)(150)}\)

\(\mathrm{= 18 \times 150}\)

\(\mathrm{= 2700}\)

leading them to think \(\mathrm{k = 2700}\). This massive number should be a red flag that something went wrong, but students often don't catch this error and end up confused or guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly convert to radians but make arithmetic errors when simplifying the final calculation.

For example, they might incorrectly calculate \(\mathrm{18 \times \frac{5\pi}{6}}\) as \(\mathrm{\frac{15\pi}{6}}\) instead of \(\mathrm{\frac{90\pi}{6}}\), or forget to reduce \(\mathrm{\frac{90\pi}{6}}\) to \(\mathrm{15\pi}\). This leads to wrong values of \(\mathrm{k}\) and confusion about the final answer.

The Bottom Line:

This problem tests whether students understand that trigonometric formulas involving angles typically require radians, not degrees. The conversion step is crucial and easy to forget under time pressure.