Circle A has a diameter that is 14 times the radius of circle B. The area of circle A is...

1. TRANSLATE the problem information

• Given information:
  • Circle A's diameter = \(14 \times (\mathrm{radius\ of\ circle\ B})\)
  • Need to find \(k\) where \(\mathrm{Area_A} = k \times \mathrm{Area_B}\)

2. INFER the approach needed

• To compare areas, I need both radii first
• Let \(r\) = radius of circle B
• Since diameter of A = \(14r\), then radius of A = \(14r \div 2 = 7r\)

3. Apply the area formula to both circles

• Area of circle A = \(\pi(7r)^2\)

\(= \pi(49r^2)\)

\(= 49\pi r^2\)

• Area of circle B = \(\pi r^2\)

4. SIMPLIFY to find the ratio k

• \(k = \frac{\mathrm{Area\ of\ circle\ A}}{\mathrm{Area\ of\ circle\ B}}\)

\(k = \frac{49\pi r^2}{\pi r^2}\)

\(= 49\)

Answer: 49

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might incorrectly interpret "diameter that is 14 times the radius" and think the radius of circle A is \(14r\) instead of \(7r\).

When they use radius A = \(14r\), they get Area A = \(\pi(14r)^2\)

\(= 196\pi r^2\), leading to \(k = 196\).

This may lead them to select Choice E (196).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly find that radius A = \(7r\) but make an error when squaring, calculating \((7r)^2 = 7r^2\) instead of \(49r^2\).

This gives them Area A = \(7\pi r^2\), so \(k = 7\).

This may lead them to select Choice A (7).

The Bottom Line:

This problem tests whether students can carefully track the diameter-to-radius conversion and correctly apply the area formula. The key insight is recognizing that diameter = \(14r\) means radius = \(7r\), not \(14r\).