The circumference of Circle A is 23 times the circumference of Circle B. The area of Circle A is k...

1. TRANSLATE the problem information

• Given information:
  • Circumference of Circle A = 23 × Circumference of Circle B
  • Area of Circle A = k × Area of Circle B
  • Need to find: value of k

2. INFER the connection strategy

• To connect circumference and area, we need to work through radius
• Both circumference and area formulas involve radius, so if we can find how the radii relate, we can find how the areas relate

3. TRANSLATE circumference relationship to radius relationship

• Using \(\mathrm{C = 2\pi r}\):
  • \(\mathrm{C_A = 2\pi r_A}\) and \(\mathrm{C_B = 2\pi r_B}\)
  • Since \(\mathrm{C_A = 23C_B}\): \(\mathrm{2\pi r_A = 23 \times 2\pi r_B}\)
  • The \(\mathrm{2\pi}\) cancels: \(\mathrm{r_A = 23r_B}\)

4. INFER the area relationship from radius relationship

• Since area \(\mathrm{A = \pi r^2}\), and we know \(\mathrm{r_A = 23r_B}\):
  • \(\mathrm{A_A = \pi(r_A)^2}\)
    \(\mathrm{A_A = \pi(23r_B)^2}\)
    \(\mathrm{A_A = \pi \times 23^2 \times r_B^2}\)
  • \(\mathrm{A_B = \pi(r_B)^2}\)

5. SIMPLIFY to find k

• \(\mathrm{k = A_A/A_B}\)
  \(\mathrm{k = (\pi \times 23^2 \times r_B^2)/(\pi \times r_B^2)}\)
• The \(\mathrm{\pi}\) and \(\mathrm{r_B^2}\) terms cancel: \(\mathrm{k = 23^2 = 529}\)

Answer: D) 529

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that area scales by the square of the linear scale factor

Many students see that the circumference is 23 times larger and assume the area is also just 23 times larger. They miss the crucial insight that when linear dimensions scale by factor k, areas scale by k². This leads them to select Choice A (23).

Second Most Common Error:

Poor TRANSLATE execution: Students confuse the relationship direction or make algebraic errors

Some students correctly start with the circumference relationship but make errors when setting up equations, such as writing \(\mathrm{r_A = 23r_B}\) as \(\mathrm{r_B = 23r_A}\), or they make arithmetic mistakes when computing \(\mathrm{23^2}\). This can lead to selecting Choice B (46) or getting confused and guessing.

The Bottom Line:

This problem tests whether students understand that geometric scaling relationships are different for linear measurements (like circumference) versus area measurements. The key insight is that area scaling follows the square of linear scaling.