The temperature T of a heated object after n minutes in a room is modeled by the equation \(\mathrm{T =...

1. TRANSLATE the problem requirements

• Given equation: \(\mathrm{T = T_{room} + (T_{initial} - T_{room}) \cdot s^n}\)
• Physical requirement: Object cools toward room temperature over time
• What this means: Temperature difference \(\mathrm{|T - T_{room}|}\) must decrease as time n increases

2. INFER the mathematical requirement

• From the equation: \(\mathrm{T - T_{room} = (T_{initial} - T_{room}) \cdot s^n}\)
• So the temperature difference depends on \(\mathrm{s^n}\)
• For cooling: We need \(\mathrm{|s^n|}\) to get smaller as n increases
• This happens when \(\mathrm{|s| \lt 1}\)

3. APPLY CONSTRAINTS to eliminate impossible choices

• Choices (A) \(\mathrm{s \lt -1}\) and (D) \(\mathrm{s \gt 1}\) both have \(\mathrm{|s| \gt 1}\)
• When \(\mathrm{|s| \gt 1}\), the value \(\mathrm{|s^n|}\) grows instead of shrinks
• Eliminate (A) and (D)

4. INFER realistic physical behavior

• Remaining choices: (B) \(\mathrm{-1 \lt s \lt 0}\) and (C) \(\mathrm{0 \lt s \lt 1}\)
• If s is negative, then \(\mathrm{s^n}\) alternates signs:
  • \(\mathrm{s^2}\) is positive, \(\mathrm{s^3}\) is negative, \(\mathrm{s^4}\) is positive...
  • Temperature would oscillate above and below room temperature
• Real cooling should be smooth, not oscillating
• Therefore s must be positive

5. APPLY CONSTRAINTS for final answer

• We need \(\mathrm{|s| \lt 1}\) AND \(\mathrm{s \gt 0}\)
• This gives us \(\mathrm{0 \lt s \lt 1}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students recognize that \(\mathrm{|s| \lt 1}\) is needed but don't consider the physical implications of negative s values.

They correctly eliminate choices (A) and (D) but then randomly choose between (B) and (C) without thinking about what negative s means physically. The oscillating temperature behavior seems abstract, so they might think "mathematically both work" and guess.

This may lead them to select Choice B (\(\mathrm{-1 \lt s \lt 0}\)).

Second Most Common Error:

Incomplete INFER analysis: Students might think that any s with \(\mathrm{|s| \gt 1}\) works, misunderstanding which direction the inequality should go.

They reason "for cooling, we need the exponential term to get bigger over time" instead of realizing it needs to get smaller to approach room temperature. This fundamental misunderstanding of the cooling process leads to the wrong mathematical requirement.

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

The Bottom Line:

This problem requires connecting abstract mathematical behavior (how exponential functions behave) with physical intuition (how objects actually cool). Students who focus only on the math or only on the physics miss the complete picture.