H = 120p + 60 The Karvonen formula above shows the relationship between Alice's target heart rate H, in beats...

1. TRANSLATE the problem information

• Given information:
  • Heart rate formula: \(\mathrm{H = 120p + 60}\)
  • Training recommendation: p should be between 0.5 and 0.85
  • Need to find: Range of H values for training

• What this tells us: We need H values when \(\mathrm{0.5 \leq p \leq 0.85}\)

2. INFER the approach

• To find the range of H values, we need to evaluate H at both endpoints of the p interval
• Since \(\mathrm{H = 120p + 60}\) is a linear function with positive slope, H increases as p increases
• Therefore, the minimum H occurs at \(\mathrm{p = 0.5}\) and maximum H occurs at \(\mathrm{p = 0.85}\)

3. SIMPLIFY to find the minimum heart rate

• When \(\mathrm{p = 0.5}\):
  \(\mathrm{H = 120(0.5) + 60}\)
  \(\mathrm{H = 60 + 60}\)
  \(\mathrm{H = 120}\)

4. SIMPLIFY to find the maximum heart rate

• When \(\mathrm{p = 0.85}\):
  \(\mathrm{H = 120(0.85) + 60}\)
  \(\mathrm{H = 102 + 60}\)
  \(\mathrm{H = 162}\)

5. TRANSLATE back to inequality notation

• Alice's target training heart rate range: \(\mathrm{120 \leq H \leq 162}\)

Answer: A. \(\mathrm{120 \leq H \leq 162}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "p be between 0.5 and 0.85" as meaning they should find what p values correspond to the given H ranges in the answer choices, rather than finding H values for the given p range.

This backward approach leads to confusion about what they're solving for, causing them to get stuck and randomly select an answer.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when calculating \(\mathrm{H = 120(0.85) + 60}\), getting \(\mathrm{120(0.85) = 120}\) instead of 102, leading to \(\mathrm{H = 180}\) instead of 162.

This calculation error might lead them to select Choice C (\(\mathrm{60 \leq H \leq 162}\)) if they correctly find the lower bound but miss that their upper bound is wrong.

The Bottom Line:

This problem requires careful translation of the constraint on p into a mathematical inequality, then systematic evaluation of the linear function at the boundary points. Students who rush through the translation step or make basic arithmetic errors will miss the correct answer range.