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 of medication concentration values at different times
  • Need to find exponential function \(\mathrm{C(t) = a \cdot b^t}\)
  • \(\mathrm{t \geq 0}\) (time cannot be negative in this context)

2. INFER the approach

• Strategy: Use the structure of exponential functions where a is the initial value and b determines growth/decay
• Start with \(\mathrm{t = 0}\) to find the initial value, then use another point to find the decay factor

3. TRANSLATE the initial condition

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

4. APPLY CONSTRAINTS to eliminate wrong answers

• Since \(\mathrm{a = 25}\), choice A is eliminated (it has \(\mathrm{a = 16}\))
• Remaining choices: B, C, and D all have \(\mathrm{a = 25}\)

5. TRANSLATE another data point to find b

• At \(\mathrm{t = 2}\): \(\mathrm{C(2) = 16.0}\)
• Substitute into \(\mathrm{C(t) = 25 \cdot b^t}\): \(\mathrm{16 = 25 \cdot b^2}\)

6. SIMPLIFY to solve for b

• Divide both sides by 25: \(\mathrm{b^2 = 16/25 = 0.64}\)
• Take square root: \(\mathrm{b = \sqrt{0.64} = 0.8}\)

7. INFER the final function and verify

• Function: \(\mathrm{C(t) = 25(0.8)^t}\)
• Check with \(\mathrm{t = 4}\): \(\mathrm{C(4) = 25(0.8)^4 = 25(0.4096) = 10.24}\) ✓ (use calculator)
• This matches the table value perfectly

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

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly set up the exponential equation or confuse which value represents the initial amount versus the decay factor.

Some students might think the initial value should be 16 because it appears in multiple answer choices, leading them to select Choice A (\(\mathrm{C(t) = 16(0.8)^t}\)) without properly checking that \(\mathrm{a = C(0) = 25}\).

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when solving \(\mathrm{16 = 25 \cdot b^2}\), particularly with fraction operations or square roots.

They might incorrectly calculate \(\mathrm{b^2 = 16/25}\) or make errors taking the square root, potentially leading them to select Choice B (\(\mathrm{C(t) = 25(0.75)^t}\)) if they arrive at \(\mathrm{b = 0.75}\) through calculation mistakes.

The Bottom Line:

This problem tests whether students can systematically extract parameters from data points and work with exponential functions methodically. The key is recognizing that \(\mathrm{t = 0}\) immediately gives the initial value, then using algebraic manipulation to find the decay factor.