A scientist initially measures 12,000 bacteria in a growth medium. 4 hours later, the scientist measures 24,000 bacteria. Assuming exponential...

1. TRANSLATE the problem information

• Given information:
  • Initial bacteria: 12,000 at time \(\mathrm{t = 0}\)
  • Later measurement: 24,000 bacteria at time \(\mathrm{t = 4}\) hours
  • Formula: \(\mathrm{P = C(2)^{rt}}\) where P is bacteria count, t is time in hours

2. INFER the solution strategy

• We have two data points and need to find the growth rate constant r
• Strategy: Use the first data point to find C, then use the second to find r
• This works because we have two unknowns (C and r) and two equations

3. SIMPLIFY to find the constant C

• At \(\mathrm{t = 0}\): \(\mathrm{P = C(2)^{r·0}}\)
  \(\mathrm{= C(2)^0}\)
  \(\mathrm{= C(1)}\)
  \(\mathrm{= C}\)
• Since \(\mathrm{P = 12,000}\) when \(\mathrm{t = 0}\): \(\mathrm{C = 12,000}\)

4. TRANSLATE the second measurement into an equation

• At \(\mathrm{t = 4}\): \(\mathrm{P = 24,000}\)
• Substitute into formula: \(\mathrm{24,000 = 12,000(2)^{4r}}\)

5. SIMPLIFY the exponential equation

• Divide both sides by 12,000: \(\mathrm{2 = (2)^{4r}}\)
• Rewrite left side as a power of 2: \(\mathrm{2^1 = 2^{4r}}\)
• Since bases are equal, exponents must be equal: \(\mathrm{1 = 4r}\)
• Solve for r: \(\mathrm{r = \frac{1}{4}}\)

Answer: B. 1/4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to solve for both C and r simultaneously from just one equation, not recognizing they need to use the initial condition (\(\mathrm{t = 0}\)) strategically.

They might try to work directly with \(\mathrm{24,000 = C(2)^{4r}}\) without first finding C, leading to confusion about how to separate two unknowns. This leads to abandoning systematic solution and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{24,000 = 12,000(2)^{4r}}\) and get to \(\mathrm{2 = 2^{4r}}\), but then incorrectly solve the exponential equation.

They might think \(\mathrm{2 = 2^{4r}}\) means \(\mathrm{4r = 2}\), forgetting that \(\mathrm{2 = 2^1}\). This leads them to select Choice C (4) instead of the correct 1/4.

The Bottom Line:

This problem tests whether students can systematically use multiple data points in exponential functions and properly manipulate exponential equations. The key insight is using the initial condition strategically to eliminate one unknown before solving for the other.