A laboratory stores a chemical solution that cools over time. The temperature of the solution decreases by 20% every 3...

1. TRANSLATE the problem information

• Given information:
  • Initial temperature: \(\mathrm{75°F}\) at time \(\mathrm{t = 0}\)
  • Temperature decreases by 20% every 3 hours
  • Target temperature: \(\mathrm{30°F}\)
  • Need equation to find time t

2. TRANSLATE the decay rate

• "Decreases by 20% every 3 hours" means:
  • The solution retains \(\mathrm{100\% - 20\% = 80\%}\) of its temperature
  • Decay factor = \(\mathrm{0.8}\) per 3-hour period

3. INFER the time interval structure

• After t hours, how many 3-hour periods have passed?
  • Number of 3-hour periods = \(\mathrm{t ÷ 3 = t/3}\)
  • So the exponent should be \(\mathrm{t/3}\), not just t

4. TRANSLATE into exponential equation

• General exponential decay: Final = Initial × (decay factor)^(number of periods)
• Our equation: \(\mathrm{T(t) = 75(0.8)^{(t/3)}}\)
• To find when \(\mathrm{T = 30}\): \(\mathrm{30 = 75(0.8)^{(t/3)}}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students correctly identify the \(\mathrm{0.8}\) decay factor but use \(\mathrm{(0.8)^t}\) instead of \(\mathrm{(0.8)^{(t/3)}}\). They think the 20% decrease happens every hour rather than every 3 hours.

Their reasoning: "It decreases by 20% over time, so after t hours it should be \(\mathrm{75(0.8)^t}\)"

This leads them to select Choice A (\(\mathrm{30 = 75(0.8)^t}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret "decreases by 20%" as meaning the temperature becomes 120% of its original value, using \(\mathrm{1.2}\) instead of \(\mathrm{0.8}\) as the multiplier.

Their reasoning: "20% change means multiply by 1.2"

This leads them to select Choice B (\(\mathrm{30 = 75(1.2)^{(t/3)}}\))

The Bottom Line:

This problem requires precise translation of both the percentage change (decrease means multiply by \(\mathrm{0.8}\), not \(\mathrm{1.2}\)) AND the time intervals (every 3 hours means the exponent is \(\mathrm{t/3}\), not t). Students who rush through the setup often miss these crucial details.