The amount A, in milligrams, of a certain medication in a patient's bloodstream t hours after administration is modeled by...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(\mathrm{A = A_0(b)^t}\) where \(\mathrm{A}\) is medication amount at time \(\mathrm{t}\)
  • \(\mathrm{A_0}\) is initial dose, \(\mathrm{b}\) is a constant
  • "Amount decreases by a fixed fraction each hour"

2. INFER what exponential decay means

• When we have \(\mathrm{A = A_0(b)^t}\) and the amount decreases over time:
  • At \(\mathrm{t = 0}\): Amount = \(\mathrm{A_0}\)
  • At \(\mathrm{t = 1}\): Amount = \(\mathrm{A_0b}\)
  • At \(\mathrm{t = 2}\): Amount = \(\mathrm{A_0b^2}\)

• For the amount to decrease each hour, we need: \(\mathrm{A_0b \lt A_0}\)
• This means: \(\mathrm{b \lt 1}\)

3. APPLY CONSTRAINTS from the real-world context

• Since medication concentration must always be positive:
  • \(\mathrm{A_0}\) is positive (initial dose \(\mathrm{\gt 0}\))
  • \(\mathrm{A}\) must stay positive for all \(\mathrm{t ≥ 0}\)
  • Therefore \(\mathrm{b}\) must be positive

• Combining our constraints: \(\mathrm{b \lt 1}\) AND \(\mathrm{b \gt 0}\)
• This gives us: \(\mathrm{0 \lt b \lt 1}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "decreases by a fixed fraction" to the mathematical requirement that \(\mathrm{b \lt 1}\) in exponential functions.

They might think: "It's an exponential function, so b must be greater than 1" (confusing this with exponential growth contexts they've seen before).

This may lead them to select Choice A (\(\mathrm{b \gt 1}\)).

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly identify that \(\mathrm{b \lt 1}\) for decay, but don't consider that medication amounts must remain positive.

They might think: "Since it's decreasing, b could be negative" (not realizing this would create alternating positive/negative values).

This may lead them to select Choice C (\(\mathrm{b \lt 0}\)).

The Bottom Line:

This problem requires understanding the connection between the verbal description of exponential decay and the mathematical constraints on the base parameter. Students must recognize that "decreases by a fixed fraction" translates to multiplication by a factor between 0 and 1.