Bacteria are growing in a liquid growth medium. There were 300{,000} cells per milliliter during an initial observation. The number...

1. TRANSLATE the problem information

• Given information:
  • Initial amount: 300,000 cells per milliliter
  • Growth pattern: doubles every 3 hours
  • Time elapsed: 15 hours
  • Find: number of cells after 15 hours

• What this tells us: This is exponential growth since the population multiplies by a constant factor over equal time periods.

2. INFER the mathematical approach

• Since the bacteria population doubles at regular intervals, this requires an exponential growth formula
• The general form is: \(\mathrm{y = a(b)^{(t/k)}}\) where:
  • \(\mathrm{a}\) = initial amount
  • \(\mathrm{b}\) = growth factor per time period
  • \(\mathrm{k}\) = length of each time period
  • \(\mathrm{t}\) = total elapsed time

3. TRANSLATE each piece into the formula

• \(\mathrm{a = 300,000}\) (initial observation)
• \(\mathrm{b = 2}\) (since "doubles" means multiplies by 2)
• \(\mathrm{k = 3}\) (doubles every 3 hours)
• \(\mathrm{t = 15}\) (we want the amount after 15 hours)

• Formula becomes: \(\mathrm{y = 300,000(2)^{(15/3)}}\)

4. SIMPLIFY the exponent and calculate

• First: \(\mathrm{15/3 = 5}\), so \(\mathrm{y = 300,000(2)^5}\)
• Calculate \(\mathrm{2^5 = 2 × 2 × 2 × 2 × 2 = 32}\)
• Final calculation: \(\mathrm{y = 300,000 × 32 = 9,600,000}\) (use calculator)

Answer: D. 9,600,000

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students misinterpret this as linear growth instead of exponential growth.

They might think: "If it doubles every 3 hours, then in 15 hours (which is 5 periods of 3 hours), it increases by \(\mathrm{5 × 300,000 = 1,500,000}\), giving a total of \(\mathrm{300,000 + 1,500,000 = 1,800,000}\)." This type of additive thinking ignores that each doubling applies to the current population, not the original.

This may lead them to select Choice A (1,500,000) or get confused and guess.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly recognize exponential growth but use 15 directly as the exponent instead of 15/3.

They set up: \(\mathrm{y = 300,000(2)^{15}}\), which gives an astronomically large number that doesn't match any answer choice. This causes them to get stuck and randomly select an answer.

The Bottom Line:

The key challenge is recognizing that "doubles every 3 hours" creates exponential, not linear, growth. Each doubling multiplies the current population by 2, creating the compounding effect that makes exponential growth so powerful.