Question:Circle A has a diameter of 6 centimeters.Circle B has an area of 81pi square centimeters.What is the sum of...

1. TRANSLATE the given information

• Given information:
  • Circle A has \(\mathrm{diameter = 6\,cm}\)
  • Circle B has \(\mathrm{area = 81\pi\,cm^2}\)
  • Need to find: sum of both circumferences

• What this tells us: We have different types of information for each circle, so we'll need different approaches to find each radius.

2. INFER the solution strategy

• Key insight: To find circumference \(\mathrm{(C = 2\pi r)}\), we need the radius of each circle
• For Circle A: We can convert diameter directly to radius
• For Circle B: We need to work backwards from area to find radius first
• Then calculate both circumferences and add them

3. Find Circle A's circumference

• TRANSLATE: Diameter = 6 cm means \(\mathrm{radius = 6/2 = 3\,cm}\)
• SIMPLIFY: \(\mathrm{Circumference = 2\pi r = 2\pi(3) = 6\pi\,cm}\)

4. Find Circle B's radius from its area

• TRANSLATE: "Area = 81π" means \(\mathrm{\pi r^2 = 81\pi}\)
• SIMPLIFY: Divide both sides by π: \(\mathrm{r^2 = 81}\)
• Take the square root: \(\mathrm{r = 9\,cm}\) (positive since radius must be positive)

5. Find Circle B's circumference

• SIMPLIFY: \(\mathrm{Circumference = 2\pi r = 2\pi(9) = 18\pi\,cm}\)

6. SIMPLIFY to find the final answer

• Sum = \(\mathrm{6\pi + 18\pi = 24\pi\,cm}\)

Answer: C (\(\mathrm{24\pi}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to find radius first for both circles, or they get confused about having different given information for each circle.

They might try to use diameter directly in a circumference formula, or struggle with the backwards calculation from area to radius for Circle B. This leads to calculation errors and may cause them to select Choice A (\(\mathrm{6\pi}\)) if they only calculated Circle A's circumference, or get stuck and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic mistakes when solving \(\mathrm{\pi r^2 = 81\pi}\), perhaps getting \(\mathrm{r = 81}\) instead of \(\mathrm{r = 9}\), or they make arithmetic errors when calculating \(\mathrm{2\pi(9)}\).

These calculation errors lead to incorrect circumference values that don't match any of the answer choices, causing confusion and potentially random selection of Choice B (\(\mathrm{18\pi}\)) if they calculated only Circle B correctly.

The Bottom Line:

This problem tests whether students can adapt their approach based on different given information (diameter vs. area) and successfully execute multi-step calculations involving circle formulas. The key challenge is recognizing that circumference always requires radius, regardless of what information is initially provided.