A circle has a radius of 6. A shaded sector in the circle has an area of 4pi. What fraction...

1. TRANSLATE the problem information

• Given information:
  • Circle \(\mathrm{radius = 6}\)
  • Sector \(\mathrm{area = 4\pi}\)
  • Need to find what fraction the sector is of the total circle

• This tells us we need to find: \(\mathrm{\frac{sector\ area}{total\ circle\ area}}\)

2. INFER the approach

• To find the fraction, we need both the sector area (given) and total circle area (must calculate)
• We'll use the circle area formula, then set up a ratio

3. Calculate the total area of the circle

• Using \(\mathrm{A = \pi r^2}\) with \(\mathrm{r = 6}\):
• Total area = \(\mathrm{\pi(6)^2 = 36\pi}\)

4. TRANSLATE into fraction form

• Fraction = \(\mathrm{\frac{sector\ area}{total\ area} = \frac{4\pi}{36\pi}}\)

5. SIMPLIFY the fraction

• The \(\mathrm{\pi}\) terms cancel: \(\mathrm{\frac{4\pi}{36\pi} = \frac{4}{36}}\)
• Find GCD of 4 and 36: \(\mathrm{GCD = 4}\)
• Divide both numerator and denominator by 4: \(\mathrm{\frac{4}{36} = \frac{1}{9}}\)

Answer: A) \(\mathrm{\frac{1}{9}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up \(\mathrm{\frac{4\pi}{36\pi}}\) but fail to fully reduce the fraction.

They might cancel the \(\mathrm{\pi}\) terms to get \(\mathrm{\frac{4}{36}}\), but then select this as their final answer without reducing further. Since \(\mathrm{\frac{4}{36}}\) appears as answer choice B, they select it thinking they're done. They don't recognize that \(\mathrm{\frac{4}{36} = \frac{1}{9}}\) when both numerator and denominator are divided by their greatest common divisor of 4.

This leads them to select Choice B (\(\mathrm{\frac{4}{36}}\)).

The Bottom Line:

This problem tests whether students can complete the full simplification process. The trap answer choice \(\mathrm{\frac{4}{36}}\) catches students who stop partway through the algebraic reduction, even though their mathematical setup is completely correct.