A circle has a radius of 2.1 inches. The area of the circle is bpi square inches, where b is...

1. TRANSLATE the problem information

• Given information:
  • Circle radius = 2.1 inches
  • Area is expressed as \(\mathrm{b\pi}\) square inches
  • Need to find the value of constant b

• What this tells us: We have a circle area expressed in a special form (\(\mathrm{b\pi}\)) and need to find what b equals.

2. INFER the approach

• Since we know the radius and need to find a value related to area, we should use the circle area formula
• Key insight: The area \(\mathrm{b\pi}\) must equal the actual area \(\mathrm{\pi r^2}\)
• Strategy: Calculate \(\mathrm{\pi r^2}\) and set it equal to \(\mathrm{b\pi}\), then solve for b

3. SIMPLIFY using the circle area formula

• Apply \(\mathrm{A = \pi r^2}\) with \(\mathrm{r = 2.1}\):

\(\mathrm{A = \pi(2.1)^2}\)
\(\mathrm{A = \pi(4.41)}\)
\(\mathrm{A = 4.41\pi}\)

4. INFER the final equation and solve

• Since the area equals both \(\mathrm{4.41\pi}\) and \(\mathrm{b\pi}\):

\(\mathrm{b\pi = 4.41\pi}\)

• Divide both sides by \(\mathrm{\pi}\):

\(\mathrm{b = 4.41}\)

Answer: 4.41

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that "\(\mathrm{b\pi}\) square inches" represents the area of the circle, instead treating b as some separate quantity unrelated to the area formula.

This leads to confusion about how to connect the given radius with the expression \(\mathrm{b\pi}\), causing them to get stuck and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{b\pi = \pi(2.1)^2}\) but make arithmetic errors when calculating \(\mathrm{(2.1)^2}\) or forget to properly isolate b.

Common calculation mistake: \(\mathrm{(2.1)^2 = 2.1 \times 2 = 4.2}\) instead of \(\mathrm{2.1 \times 2.1 = 4.41}\), leading to an incorrect final answer.

The Bottom Line:

This problem tests whether students can connect a geometric formula with an algebraic expression. The key challenge is recognizing that \(\mathrm{b\pi}\) isn't just notation—it's literally the area of the circle expressed in coefficient form.