A biologist is studying a culture of bacteria. The initial population of the culture was 2,000 bacteria. Due to the...

1. TRANSLATE the problem information

• Given information:
  • Initial population: 2,000 bacteria
  • Population reduces by 25% every 4 hours
  • Need to find \(\mathrm{B(t)}\) = population after \(\mathrm{t}\) hours
• What this tells us: We have an exponential decay situation with a specific time period

2. INFER the exponential decay model structure

• For exponential decay problems, we use: \(\mathrm{F(t) = P \times b^{(t/k)}}\)
• \(\mathrm{P}\) = initial amount
• \(\mathrm{b}\) = decay factor (fraction that remains after each period)
• \(\mathrm{t/k}\) = number of decay periods that occur in time \(\mathrm{t}\)

3. TRANSLATE each component from the problem

• Initial amount: \(\mathrm{P = 2000}\)
• Decay factor: If 25% is removed, then \(\mathrm{100\% - 25\% = 75\%}\) remains
  So \(\mathrm{b = 0.75}\)
• Time period: \(\mathrm{k = 4}\) hours (decay happens every 4 hours)

4. INFER the correct exponent

• Since decay happens every 4 hours, in \(\mathrm{t}\) hours there are \(\mathrm{t/4}\) periods
• So the exponent should be \(\mathrm{t/4}\)

5. Assemble the final function

• \(\mathrm{B(t) = P \times b^{(t/k)}}\)
• \(\mathrm{B(t) = 2000 \times (0.75)^{(t/4)}}\)

Answer: B. \(\mathrm{B(t) = 2000(0.75)^{(t/4)}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the decay factor by using 0.25 instead of 0.75

Students see "reduced by 25%" and directly use 0.25 as the decay factor, not realizing this represents what's lost, not what remains. They think: "25% reduction means multiply by 0.25 each period."

This may lead them to select Choice A (\(\mathrm{B(t) = 2000(0.25)^{4t}}\)) - though this also has the wrong exponent structure.

Second Most Common Error:

Poor INFER reasoning about the exponent: Students incorrectly construct the time relationship

Students might think: "Every 4 hours means multiply the time by 4" instead of recognizing that \(\mathrm{t/4}\) represents how many 4-hour periods fit into \(\mathrm{t}\) hours. They reverse the relationship.

This may lead them to select Choice C (\(\mathrm{B(t) = 2000(0.75)^{4t}}\)) - correct decay factor but wrong time structure.

The Bottom Line:

The key challenge is distinguishing between what's lost (25%) versus what remains (75%), combined with correctly modeling how time periods work in exponential functions.