On day 0, a bacterial culture contained 400 bacteria. Every 6 hours after day 0, the bacteria population decreases to...

1. TRANSLATE the problem information

• Given information:
  • Day 0: 400 bacteria
  • Every 6 hours: population decreases to 80% of what it was
• What this tells us: We multiply by \(0.8\) every 6-hour period

2. INFER the mathematical pattern

• Since "decreases to 80%" means the new amount is 80% of the old amount, we multiply by \(0.8\)
• This multiplication happens every 6 hours
• We need to figure out how many 6-hour periods occur in t hours

3. INFER the exponent structure

• After 6 hours: \(400(0.8)^1\)
• After 12 hours: \(400(0.8)^2\)
• After 18 hours: \(400(0.8)^3\)
• After t hours: We've had t/6 periods of 6 hours, so \(400(0.8)^{\mathrm{t/6}}\)

4. Match to answer choices

• Looking for: initial value 400, base 0.8, exponent t/6
• Choice D: \(\mathrm{h(t)} = 400(0.8)^{\mathrm{t/6}}\) ✓

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Confusing "decreases to 80%" with "decreases by 80%"

Students think: "If it decreases by 80%, then only 20% remains, so I multiply by 0.2"

This leads them to look for 0.2 as the base and select Choice A (\(400(0.2)^{\mathrm{t/6}}\)) or Choice B (\(400(0.2)^{6\mathrm{t}}\))

Second Most Common Error:

Poor INFER reasoning about time intervals: Getting the exponent backwards

Students think: "Every 6 hours something happens, so after t hours, it should be something with 6t"

This may lead them to select Choice C (\(400(0.8)^{6\mathrm{t}}\))

The Bottom Line:

This problem requires careful attention to language ("to 80%" vs "by 80%") and understanding how time intervals translate to exponents in exponential functions.