Question:A circular lawn has radius 5 meters. A sprinkler wets exactly a sector of the lawn with area 5pi/2 square...

1. TRANSLATE the problem information

• Given information:
  • Circular lawn with \(\mathrm{radius = 5\text{ meters}}\)
  • Sector area = \(\frac{5\pi}{2}\text{ square meters}\)
  • Need: fraction of circumference corresponding to the wetted arc

2. INFER the key relationship

• The crucial insight: For any sector, the fraction of total area occupied equals the fraction of total circumference covered
• Why? Both fractions depend on the same central angle θ:
  • Area fraction = \(\frac{\theta}{2\pi}\)
  • Arc length fraction = \(\frac{\theta}{2\pi}\)
• This means we can solve using areas instead of working with arc lengths directly

3. Calculate the total area of the lawn

Total area = \(\pi r^2 = \pi(5)^2 = 25\pi\text{ square meters}\)

4. SIMPLIFY to find the area fraction

Fraction of area wetted = \(\frac{\text{Sector area}}{\text{Total area}}\)

\(= \frac{5\pi/2}{25\pi}\)

\(= \frac{5\pi}{2} \times \frac{1}{25\pi}\)

\(= \frac{5}{2 \times 25} = \frac{5}{50} = \frac{1}{10}\)

5. Apply the key relationship

• Since area fraction = circumference fraction
• Fraction of circumference = \(\frac{1}{10}\)

Answer: \(\frac{1}{10}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the proportional relationship between area fraction and arc fraction, instead attempting to calculate arc length directly without knowing the central angle.

They might try to find the arc length using \(s = r\theta\), but realize they don't know θ, then get stuck and abandon a systematic approach. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\frac{5\pi/2}{25\pi}\) but make fraction reduction errors, such as:

• Forgetting to cancel the π terms
• Incorrectly reducing \(\frac{5}{50}\) (perhaps getting \(\frac{1}{5}\) instead of \(\frac{1}{10}\))
• Making arithmetic mistakes in the multiplication

This may lead them to select an incorrect numerical answer.

The Bottom Line:

The key breakthrough is recognizing that you don't need to calculate arc length directly—the elegant relationship between area and circumference fractions makes this a much simpler problem than it first appears.