A circular ring is formed by cutting a circular hole from the center of a larger circular disk. The outer...

1. TRANSLATE the problem information

• Given information:
  • Outer circle radius = \(\mathrm{3.2}\) inches
  • Inner circle radius = \(\mathrm{1.8}\) inches
  • Ring area = \(\mathrm{c\pi}\) square inches
  • Need to find the constant c
• What this tells us: We have a ring (annulus) where we need to find the area

2. INFER the approach

• A ring's area equals the outer circle's area minus the inner circle's area
• Once we find the total area, we can identify the value of c since area = \(\mathrm{c\pi}\)

3. SIMPLIFY to find the outer circle area

• Area of outer circle = \(\mathrm{\pi \times (radius)^2}\)
• Area = \(\mathrm{\pi \times (3.2)^2}\)
• Area = \(\mathrm{\pi \times 10.24}\)
• Area = \(\mathrm{10.24\pi}\) square inches

4. SIMPLIFY to find the inner circle area

• Area of inner circle = \(\mathrm{\pi \times (1.8)^2}\)
• Area = \(\mathrm{\pi \times 3.24}\)
• Area = \(\mathrm{3.24\pi}\) square inches

5. SIMPLIFY to find the ring area

• Ring area = Outer area - Inner area
• Ring area = \(\mathrm{10.24\pi - 3.24\pi}\)
• Ring area = \(\mathrm{7\pi}\) square inches

6. INFER the value of c

• Since the ring area = \(\mathrm{c\pi}\) and we found the area = \(\mathrm{7\pi}\)
• Therefore: \(\mathrm{c\pi = 7\pi}\), which means \(\mathrm{c = 7}\)

Answer: 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misunderstand what a 'ring' means and try to add the areas instead of subtracting them, or they might forget that the inner circle represents a hole that must be removed from the total area.

This leads them to calculate \(\mathrm{10.24\pi + 3.24\pi = 13.24\pi}\), giving \(\mathrm{c = 13.24}\), which doesn't match the student response format and causes confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly understand they need to subtract areas but make arithmetic errors when calculating \(\mathrm{(3.2)^2}\) or \(\mathrm{(1.8)^2}\), leading to incorrect intermediate values.

For example, miscalculating \(\mathrm{3.2^2}\) as \(\mathrm{9.64}\) instead of \(\mathrm{10.24}\) would lead to a ring area of \(\mathrm{6.4\pi}\) and \(\mathrm{c = 6.4}\), requiring rounding to \(\mathrm{6}\).

The Bottom Line:

This problem tests whether students can visualize what a ring represents geometrically and correctly apply the area subtraction concept, combined with careful arithmetic execution of the squared terms.