A cup of coffee cools down such that each minute, its temperature above room temperature decreases by 3.2% of the...

1. TRANSLATE the problem information

• Given information:
  • Coffee temperature above room temperature decreases each minute
  • Each minute it decreases by 3.2% of the current temperature difference
  • Need to identify the function type

• What this tells us: The word "current" is crucial - this means the decrease amount changes each minute based on how much temperature difference remains

2. INFER what type of change this represents

• Key insight: When something changes by a fixed percentage of its current value, this creates exponential behavior
• This is different from changing by a fixed amount each time (which would be linear)

3. Set up the mathematical model

• Let \(\mathrm{T(t)}\) = temperature above room temperature at time t
• After 1 minute: \(\mathrm{T(1) = T_0 - 0.032T_0}\)
  \(\mathrm{= T_0(1 - 0.032)}\)
  \(\mathrm{= 0.968T_0}\)
• After t minutes: \(\mathrm{T(t) = T_0(0.968)^t}\)

4. INFER whether this is increasing or decreasing

• Since \(\mathrm{0.968 \lt 1}\), each multiplication makes the value smaller
• Therefore this is a decreasing exponential function

Answer: (A) Decreasing exponential

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret "decreases by 3.2%" as meaning the temperature drops by the same fixed amount each minute, rather than understanding that 3.2% of the current temperature means the actual decrease amount gets smaller over time.

This leads them to think: "If it's decreasing by the same amount each time, that's linear," causing them to select Choice (B) (Decreasing linear).

Second Most Common Error:

Missing conceptual knowledge about exponential vs linear patterns: Students might recognize that percentages are involved but still confuse the fundamental difference between constant rate of change (linear) versus constant rate of proportional change (exponential).

This leads to uncertainty about whether percentage changes create linear or exponential relationships, causing them to get stuck and guess between the decreasing options.

The Bottom Line:

The key distinction is recognizing that "3.2% of the current temperature" means the actual amount of decrease gets smaller each minute as the temperature difference shrinks, which is the hallmark of exponential decay rather than linear change.