\(\mathrm{b(x) = 12,000(0.6)^{x/12}}\) The function b gives the number of bacteria in a culture after x hours of observation. If...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{b(x) = 12{,}000(0.6)^{(x/12)}}\) where \(\mathrm{x}\) = hours
  • Need to find p% daily decrease
  • There are 24 hours in one day

2. INFER the approach needed

• To find the daily percentage change, we need to see what happens to the bacteria count over exactly 24 hours
• We'll compare the multiplier over one day to determine the percentage decrease

3. SIMPLIFY the exponential expression for one day

• Substitute \(\mathrm{x = 24}\) hours into the exponent:

\(\mathrm{(0.6)^{(24/12)} = (0.6)^2 = 0.36}\)

• This means each day the bacteria count gets multiplied by 0.36

4. TRANSLATE the multiplier to percentage change

• Multiplier of 0.36 means the count becomes 36% of the previous day's amount
• If it becomes 36%, then it decreased by: \(\mathrm{100\% - 36\% = 64\%}\)

Answer: D (64)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what "decreases by p%" means. They might think the 0.6 base directly represents the percentage decrease, leading them to select Choice A (6) or confuse 0.6 with 60% decrease and select an answer not even listed.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize they need to evaluate the function over exactly 24 hours. They might use \(\mathrm{x = 12}\) or just work with the base 0.6 directly, leading to incorrect calculations and potentially selecting Choice B (36) by confusing the remaining percentage with the decrease percentage.

The Bottom Line:

This problem requires clear thinking about what exponential decay means over a specific time period. The key insight is that you must evaluate the function over exactly one day (24 hours) to find the daily change rate, then properly convert from "what remains" to "what decreased."