Circle P has an area that is 17,850 times the area of Circle Q. The circumference of Circle P is...

1. TRANSLATE the problem information

• Given information:
  • Area of circle P = 17,850 × Area of circle Q
  • Circumference of circle P = m × Circumference of circle Q
  • Find the value of m

2. TRANSLATE to mathematical relationships

• Let \(\mathrm{r_P}\) = radius of circle P, \(\mathrm{r_Q}\) = radius of circle Q
• Area relationship: \(\mathrm{\pi(r_P)^2 = 17{,}850 \times \pi(r_Q)^2}\)
• Circumference relationship: \(\mathrm{2\pi(r_P) = m \times 2\pi(r_Q)}\)

3. SIMPLIFY the area relationship

• Divide both sides by \(\mathrm{\pi}\): \(\mathrm{(r_P)^2 = 17{,}850 \times (r_Q)^2}\)
• Rearrange: \(\mathrm{(r_P/r_Q)^2 = 17{,}850}\)
• Take square root: \(\mathrm{r_P/r_Q = \sqrt{17{,}850}}\)

4. INFER the circumference relationship

• From \(\mathrm{C_P = m \times C_Q}\): \(\mathrm{2\pi(r_P) = m \times 2\pi(r_Q)}\)
• SIMPLIFY: Divide by \(\mathrm{2\pi}\): \(\mathrm{r_P = m \times r_Q}\)
• Therefore: \(\mathrm{m = r_P/r_Q}\)

5. INFER the final connection

• We found \(\mathrm{r_P/r_Q = \sqrt{17{,}850}}\)
• We found \(\mathrm{m = r_P/r_Q}\)
• Therefore: \(\mathrm{m = \sqrt{17{,}850}}\)

6. SIMPLIFY the square root calculation

• Use calculator: \(\mathrm{\sqrt{17{,}850} \approx 133.6}\)
• Can verify: \(\mathrm{133^2 = 17{,}689}\), \(\mathrm{134^2 = 17{,}956}\) (17,850 falls between)

Answer: 133.6 (or approximately 134)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the connection between area ratios and circumference ratios through the radius relationship.

They might try to directly relate areas to circumferences without understanding that both depend on radius. This leads to confusion about how to use the given area ratio of 17,850 to find the circumference ratio m. Students may attempt incorrect approaches like taking √17,850 and then squaring it again, or trying to find some direct area-to-circumference conversion formula.

This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make calculation errors when finding √17,850.

Without a calculator, this requires estimation skills that many students lack. They might estimate incorrectly (perhaps thinking it's around 100 or 200), or make arithmetic errors when checking perfect squares. Even with calculators, students sometimes input the calculation wrong or misread the display.

This may lead them to select an incorrect numerical answer.

The Bottom Line:

This problem challenges students to see the elegant relationship between area scaling and circumference scaling through radius. The key insight is that when areas differ by a factor of k, the radii (and therefore circumferences) differ by a factor of √k.