Question:A scientist is studying a radioactive isotope. The mass of the isotope, \(\mathrm{M(t)}\) in grams, remaining after t hours can...

1. TRANSLATE the problem information

• Given information:
  • At \(\mathrm{t = 0}\) hours: mass = 64 grams
  • At \(\mathrm{t = 2}\) hours: mass = 4 grams
  • Need exponential function model

• What this tells us: We have two data points to determine our exponential function parameters

2. INFER the approach

• Since this is exponential decay, we use the form: \(\mathrm{M(t) = M_0 \cdot r^t}\)
• Strategy: Use the two given points to find \(\mathrm{M_0}\) (initial amount) and \(\mathrm{r}\) (decay rate)
• Start with \(\mathrm{t = 0}\) since it directly gives us \(\mathrm{M_0}\)

3. TRANSLATE the initial condition

• At \(\mathrm{t = 0}\): \(\mathrm{M(0) = 64}\)
• Substituting: \(\mathrm{M(0) = M_0 \cdot r^0 = M_0 \cdot 1 = M_0}\)
• Therefore: \(\mathrm{M_0 = 64}\)

4. TRANSLATE and SIMPLIFY using the second condition

• At \(\mathrm{t = 2}\): \(\mathrm{M(2) = 4}\)
• Substituting into \(\mathrm{M(t) = 64 \cdot r^t}\): \(\mathrm{64 \cdot r^2 = 4}\)
• SIMPLIFY to find \(\mathrm{r}\):
  • Divide both sides by 64: \(\mathrm{r^2 = \frac{4}{64} = \frac{1}{16}}\)
  • Take square root: \(\mathrm{r = \sqrt{\frac{1}{16}} = \frac{1}{4}}\)

5. INFER the complete model

• With \(\mathrm{M_0 = 64}\) and \(\mathrm{r = \frac{1}{4}}\), our function is: \(\mathrm{M(t) = 64 \cdot (\frac{1}{4})^t}\)
• This matches choice C

Answer: C. \(\mathrm{M(t) = 64 \cdot (\frac{1}{4})^t}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize this as an exponential model and instead assume linear decay.

Reasoning: "The mass decreased from 64 to 4 grams in 2 hours, so it loses 30 grams every hour." This leads to a linear function like \(\mathrm{M(t) = 64 - 30t}\).

This may lead them to select Choice A (\(\mathrm{M(t) = 64 - 30t}\))

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{64 \cdot r^2 = 4}\) but make calculation errors when finding \(\mathrm{r}\).

They might incorrectly think \(\mathrm{r^2 = \frac{4}{64}}\) gives \(\mathrm{r = \frac{1}{16}}\) (forgetting to take the square root), or they might compute the fraction incorrectly.

This may lead them to select Choice B (\(\mathrm{M(t) = 64 \cdot (\frac{1}{16})^t}\))

The Bottom Line:

This problem requires recognizing exponential patterns in real-world contexts and carefully executing algebraic steps. The key insight is that exponential decay means the rate of decrease is proportional to the current amount, not constant over time.