A laboratory models the decay of a radioactive substance by assuming the mass halves every 6 years. In 2001, a...

1. TRANSLATE the problem information

• Given information:
  • 2001: sample has \(48\) grams
  • Mass halves every \(6\) years
  • Need to find when mass is FIRST less than \(12\) grams

• What this tells us: This is exponential decay where we multiply by \(\frac{1}{2}\) every \(6\) years

2. INFER the approach

• We need to track the mass at each answer choice year
• Each 6-year period: new mass = current mass × \(\frac{1}{2}\)
• We stop when we first get below \(12\) grams (not equal to \(12\))

3. SIMPLIFY by calculating each time period

• After 1 half-life (2001 + 6 = 2007): \(48 \times \frac{1}{2} = 24\) grams
• After 2 half-lives (2001 + 12 = 2013): \(48 \times \frac{1}{4} = 12\) grams
• After 3 half-lives (2001 + 18 = 2019): \(48 \times \frac{1}{8} = 6\) grams

4. APPLY CONSTRAINTS to identify the correct answer

• 2007: \(24\) grams (still greater than \(12\))
• 2013: \(12\) grams (equals \(12\), but we need LESS than \(12\))
• 2019: \(6\) grams (first time less than \(12\) ✓)

Answer: C) 2019

Why Students Usually Falter on This Problem

Most Common Error Path:

APPLY CONSTRAINTS: Missing the distinction between "less than" and "less than or equal to"

Students correctly calculate that the mass equals \(12\) grams in 2013, but fail to recognize that the question asks for when it's "first less than" \(12\) grams, not "first equal to or less than" \(12\) grams.

This leads them to select Choice B (2013).

Second Most Common Error:

TRANSLATE: Misunderstanding the half-life timing

Students might think "halves every 6 years" means it takes 6 years from any starting mass to reach half that amount, rather than understanding that each 6-year interval from 2001 represents one half-life period.

This causes confusion about which years to check and may lead to guessing.

The Bottom Line:

This problem tests careful attention to inequality language ("less than" vs "equal to") combined with systematic application of exponential decay. Students who rush through the constraint checking often miss the precise wording requirement.