Point A is located at coordinates \((5.6, 8.2)\) and point B is located at coordinates \((5.6, \mathrm{b})\), where b is...

1. TRANSLATE the problem information

• Given information:
  • Point A: \((5.6, 8.2)\)
  • Point B: \((5.6, \mathrm{b})\) where b is positive
  • Distance between A and B: 24.6 units

• What this tells us: Both points have the same x-coordinate, so they lie on a vertical line

2. INFER the appropriate distance approach

• Since points are on a vertical line, we don't need the full distance formula
• Distance between points on a vertical line = absolute difference of y-coordinates
• We need: \(|\mathrm{b} - 8.2| = 24.6\)

3. CONSIDER ALL CASES for the absolute value equation

• \(|\mathrm{b} - 8.2| = 24.6\) means either:
  • Case 1: \(\mathrm{b} - 8.2 = 24.6\)
  • Case 2: \(\mathrm{b} - 8.2 = -24.6\)

4. SIMPLIFY each case

• Case 1: \(\mathrm{b} - 8.2 = 24.6\) → \(\mathrm{b} = 24.6 + 8.2 = 32.8\)
• Case 2: \(\mathrm{b} - 8.2 = -24.6\) → \(\mathrm{b} = -24.6 + 8.2 = -16.4\)

5. APPLY CONSTRAINTS to select final answer

• Since the problem states that b is positive:
  • \(\mathrm{b} = 32.8\) ✓ (positive)
  • \(\mathrm{b} = -16.4\) ✗ (negative, violates constraint)

Answer: D (32.8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES reasoning: Students solve only one case of the absolute value equation, typically \(\mathrm{b} - 8.2 = 24.6\), getting \(\mathrm{b} = 32.8\), but they don't consider the second case where \(\mathrm{b} - 8.2 = -24.6\).

While this actually leads to the correct answer in this problem, the incomplete reasoning shows a fundamental gap in understanding absolute value equations. In different problems, this oversight could lead to missing the correct answer entirely.

Second Most Common Error:

Poor TRANSLATE execution: Students attempt to use the full distance formula \(\sqrt{(\mathrm{x_2}-\mathrm{x_1})^2 + (\mathrm{y_2}-\mathrm{y_1})^2}\) instead of recognizing the vertical line simplification. This leads to: \(\sqrt{(5.6-5.6)^2 + (\mathrm{b}-8.2)^2} = \sqrt{0 + (\mathrm{b}-8.2)^2} = |\mathrm{b}-8.2|\), which eventually gives the same equation but wastes time and increases chances for calculation errors.

This may lead to confusion and potentially abandoning the systematic approach for guessing.

The Bottom Line:

This problem tests whether students can recognize geometric simplifications (vertical line case) and properly handle absolute value equations with constraints. Success requires both geometric insight and algebraic completeness.