Question:A scientist measures the mass of a sample of a radioactive substance. The table below shows the exponential relationship between...

1. TRANSLATE the problem information

• Given information:
  • Table shows time t (days) vs mass m (mg)
  • \(\mathrm{t = 0: m = 800.0\ mg}\)
  • \(\mathrm{t = 1: m = 788.0\ mg}\)
  • \(\mathrm{t = 2: m = 776.18\ mg}\)
  • Problem states this is an exponential relationship
• This tells us we need to find an equation of the form \(\mathrm{m = a(b)^t}\)

2. INFER the initial value and pattern

• From the table, when \(\mathrm{t = 0, m = 800.0\ mg}\)
• This means the initial value \(\mathrm{a = 800}\)
• The mass is decreasing over time (800 → 788 → 776.18), so this is exponential decay
• This eliminates choice (D) immediately since it's missing the initial value 800

3. INFER the decay factor

• For exponential decay, we need a factor less than 1
• Calculate the ratio from day 0 to day 1: \(\mathrm{788.0 ÷ 800.0 = 0.985}\)
• Notice that \(\mathrm{0.985 = 1 - 0.015}\)
• So the decay factor is \(\mathrm{(1 - 0.015)}\)

4. SIMPLIFY to verify our model

• Our proposed equation: \(\mathrm{m = 800(1 - 0.015)^t}\)
• Check with \(\mathrm{t = 2}\):
  \(\mathrm{m = 800(0.985)^2}\)
  \(\mathrm{m = 800(0.970225)}\)
  \(\mathrm{m = 776.18}\) ✓ (use calculator)
• This matches the table data perfectly

5. INFER why other choices are wrong

• Choice (B): \(\mathrm{800(1 + 0.015)^t}\) represents growth (increasing), but our data shows decay
• Choice (C): \(\mathrm{800 - 12t}\) is linear, but the problem states exponential relationship
• Choice (D): Missing the initial mass of 800

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse exponential growth and decay patterns. They see the 0.015 rate and think "growth" because it's positive, leading them to select the \(\mathrm{(1 + 0.015)}\) form instead of recognizing that the decreasing mass values indicate decay requiring \(\mathrm{(1 - 0.015)}\).

This may lead them to select Choice B (\(\mathrm{800(1 + 0.015)^t}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Students focus on the constant decrease of "about 12 mg per day" from the first interval \(\mathrm{(800 - 788 = 12)}\) and assume this pattern continues linearly, missing that the problem explicitly states an exponential relationship and that the second interval decrease is actually 11.82 mg, not 12.

This may lead them to select Choice C (\(\mathrm{800 - 12t}\))

The Bottom Line:

The key challenge is recognizing that even though we calculate a positive rate (0.015), we use it in the decay form \(\mathrm{(1 - r)}\) because the actual mass values are decreasing. Students must connect the direction of change in the data to the correct mathematical form.