The concentration of a certain medication in a patient's bloodstream decreases over time. The table shows the concentration \(\mathrm{C(t)}\), in...

1. TRANSLATE the problem information

• Given information:
  • Table shows concentration C(t) at different times t
  • Need exponential function \(\mathrm{C(t) = a \cdot b^t}\)
  • \(\mathrm{t \geq 0}\) (time constraint)

• What this tells us: We need to find values for a (initial value) and b (decay factor)

2. INFER the approach

• Start with t = 0 since \(\mathrm{b^0 = 1}\), making it easiest to find the initial value a
• Use another data point to solve for b
• Verify with the third point to confirm our function

3. TRANSLATE the initial value

• At t = 0: \(\mathrm{C(0) = 25.0}\)
• In \(\mathrm{C(t) = a \cdot b^t}\): \(\mathrm{C(0) = a \cdot b^0 = a \cdot 1 = a}\)
• Therefore: \(\mathrm{a = 25}\)

This immediately eliminates choice A, which starts with 16.

4. SIMPLIFY to find the decay factor

• Using point \(\mathrm{(t = 2, C(t) = 16)}\):
  \(\mathrm{16 = 25 \cdot b^2}\)
• Divide both sides by 25:
  \(\mathrm{b^2 = \frac{16}{25} = 0.64}\)
• Take the square root (use calculator if needed):
  \(\mathrm{b = \sqrt{0.64} = 0.8}\)

5. INFER the complete function and verify

• Our function: \(\mathrm{C(t) = 25(0.8)^t}\)
• Check with t = 4: \(\mathrm{C(4) = 25(0.8)^4 = 25(0.4096) = 10.24}\) ✓
• This matches the table data perfectly

Answer: C. \(\mathrm{C(t) = 25(0.8)^t}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students confuse which value represents the initial amount versus other parameters in the exponential function.

Some students might think the decay factor should be calculated directly from the ratio \(\mathrm{\frac{16}{25} = 0.64}\), not recognizing that this ratio represents \(\mathrm{b^2}\) (since we're using \(\mathrm{t = 2}\)). They might select Choice B (\(\mathrm{C(t) = 25(0.75)^t}\)) thinking 0.75 seems like a reasonable decay factor.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{16 = 25 \cdot b^2}\) but make algebraic errors when solving for b.

They might incorrectly calculate \(\mathrm{b^2 = \frac{16}{25}}\) or make mistakes taking the square root, potentially leading them to guess among the remaining choices or select an incorrect decay factor.

The Bottom Line:

This problem requires students to systematically use the structure of exponential functions and carefully work with the relationship between multiple data points, not just pattern recognition from the table values.