At the start of an experiment, a bacterial culture has 120 cells. The number of cells t hours after the...

1. TRANSLATE the problem information

• Given information:
  • Original equation: \(\mathrm{N = 120(3)^{(t/5)}}\) where t is in hours
  • Need new equation where q is in minutes
  • Same bacterial growth pattern, just different time units
• This tells us we need to convert between hours and minutes

2. INFER the conversion approach

• Since the original uses hours but we want minutes, we need to express hours in terms of minutes
• Key insight: If q is minutes, then \(\mathrm{t = q/60}\) hours
• We'll substitute this relationship into the exponent

3. SIMPLIFY through substitution

• Start with: \(\mathrm{N = 120(3)^{(t/5)}}\)
• Substitute \(\mathrm{t = q/60}\): \(\mathrm{N = 120(3)^{((q/60)/5)}}\)
• Simplify the exponent: \(\mathrm{(q/60)/5 = q/(60 × 5) = q/300}\)
• Final equation: \(\mathrm{N = 120(3)^{(q/300)}}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students directly replace t with q without considering unit conversion.

They think "t represents time, q represents time, so just swap them." This ignores that t and q measure time in different units (hours vs minutes). This leads them to select Choice A (\(\mathrm{N = 120(3)^{(q/5)}}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{t = q/60}\) but make algebraic errors when simplifying the compound fraction.

They might incorrectly compute \(\mathrm{(q/60)/5}\) as \(\mathrm{q/12}\) instead of \(\mathrm{q/300}\), perhaps by only dividing the denominator by 5. This leads them to select Choice B (\(\mathrm{N = 120(3)^{(q/12)}}\)).

The Bottom Line:

This problem tests whether students understand that changing units in exponential models requires careful substitution, not just variable swapping. The key is recognizing that time units affect the rate expression in the exponent.