A model estimates that at the end of each year from 2017 through 2020, a sample's mass was 20% less...

1. TRANSLATE the problem information

• Given information:
  • Mass decreases by 20% each year from 2017-2020
  • At end of 2018, mass = 32 grams
  • \(\mathrm{t}\) = years after end of 2016
  • Need equation for mass \(\mathrm{m}\) in terms of \(\mathrm{t}\)

• What this tells us:
  • "20% less" means mass keeps 80% each year → multiplier = \(0.8\)
  • End of 2018 is \(\mathrm{t} = 2\) (two years after end of 2016)

2. INFER the model structure

• Since mass decreases by same percentage each year, this is exponential decay
• Standard form: \(\mathrm{m = m_0(0.8)^t}\) where \(\mathrm{m_0}\) is initial mass at \(\mathrm{t} = 0\)

3. INFER how to find the initial value

• Use the condition: when \(\mathrm{t} = 2\), \(\mathrm{m} = 32\)
• Substitute: \(32 = \mathrm{m_0}(0.8)^2\)

4. SIMPLIFY to find \(\mathrm{m_0}\)

• \(32 = \mathrm{m_0}(0.64)\)
• \(\mathrm{m_0} = 32 \div 0.64 = 50\)

5. INFER the complete model

• Substituting back: \(\mathrm{m} = 50(0.8)^t\)
• This matches choice (B)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students incorrectly interpret "20% less each year" as subtracting \(0.2\) instead of multiplying by \(0.8\), or they confuse which year corresponds to which t-value.

For example, they might think "20% less" means the multiplier is \(0.2\), leading to models like \(\mathrm{m} = \mathrm{m_0}(0.2)^t\). Or they might think end of 2018 corresponds to \(\mathrm{t} = 1\) instead of \(\mathrm{t} = 2\). These translation errors cause them to select wrong initial setups, making it impossible to match the given condition that \(\mathrm{m} = 32\) when \(\mathrm{t} = 2\).

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students set up the correct equation \(32 = \mathrm{m_0}(0.64)\) but make arithmetic errors when dividing, getting \(\mathrm{m_0} = 20\) instead of \(\mathrm{m_0} = 50\).

With \(\mathrm{m_0} = 20\), they would look for \(\mathrm{m} = 20(0.8)^t\), which isn't among the choices, causing them to second-guess their approach and potentially select Choice (A) (\(32(0.8)^t\)) thinking 32 should be the coefficient.

The Bottom Line:

This problem challenges students to correctly translate percentage language into mathematical multipliers and carefully track the timeline to match variables with given conditions.