A circle in the xy-plane is represented by the equation \((\mathrm{x} - 2)^2 + (\mathrm{y} + 9)^2 = 100\mathrm{m}^6\), where...

1. TRANSLATE the problem information

• Given equation: \((\mathrm{x} - 2)^2 + (\mathrm{y} + 9)^2 = 100\mathrm{m}^6\)
• Need to find: circumference of this circle

2. INFER the solution approach

• This equation matches the standard form of a circle: \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\)
• To find circumference, I need the radius first
• The right side of the equation (\(100\mathrm{m}^6\)) represents \(\mathrm{r}^2\)

3. SIMPLIFY to find the radius

• Since \(\mathrm{r}^2 = 100\mathrm{m}^6\), I need: \(\mathrm{r} = \sqrt{100\mathrm{m}^6}\)
• Break this down:
  \(\mathrm{r} = \sqrt{100} \times \sqrt{\mathrm{m}^6}\)
  \(= 10 \times \mathrm{m}^3\)
  \(= 10\mathrm{m}^3\)

4. INFER and apply the circumference formula

• Circumference formula: \(\mathrm{C} = 2\pi\mathrm{r}\)
• Substitute \(\mathrm{r} = 10\mathrm{m}^3\):
  \(\mathrm{C} = 2\pi(10\mathrm{m}^3)\)
  \(= 20\pi\mathrm{m}^3\)

Answer: B (\(20\pi\mathrm{m}^3\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make calculation errors when taking the square root of \(100\mathrm{m}^6\)

Many students correctly identify that \(\mathrm{r}^2 = 100\mathrm{m}^6\) but then calculate \(\mathrm{r} = \sqrt{100\mathrm{m}^6} = 100\mathrm{m}^3\), forgetting that \(\sqrt{\mathrm{m}^6} = \mathrm{m}^3\), not \(\mathrm{m}^6\). Using this incorrect radius in \(\mathrm{C} = 2\pi\mathrm{r}\) gives \(\mathrm{C} = 2\pi(100\mathrm{m}^3) = 200\pi\mathrm{m}^3\), which doesn't match any answer choice.

This leads to confusion and guessing among the given options.

Second Most Common Error:

Missing conceptual knowledge: Not recognizing the standard form of a circle equation

Some students don't immediately see that \((\mathrm{x} - 2)^2 + (\mathrm{y} + 9)^2 = 100\mathrm{m}^6\) represents a circle, so they get stuck on how to approach finding circumference. Without this recognition, they can't connect the equation to the radius needed for circumference calculation.

This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students can bridge equation recognition with formula application - they need to see the circle structure hidden in the algebra, then execute the square root calculation correctly.