The number of bacteria in a liquid medium doubles every day. There are 44{,}000 bacteria in the liquid medium at...

Step-by-Step Solution

Step 1: TRANSLATE the problem information

• Given information:
  • Initial bacteria count: 44,000
  • Growth pattern: population doubles every day
  • Find: equation for bacteria count y after t days

Step 2: INFER the mathematical model needed

• This describes exponential growth since the population multiplies by a constant factor each day
• Exponential functions have the form: \(\mathrm{y = a(b)^t}\)
• Where: \(\mathrm{a}\) = initial value, \(\mathrm{b}\) = growth factor, \(\mathrm{t}\) = time

Step 3: TRANSLATE the components into the formula

• Initial value: \(\mathrm{a = 44,000}\) (bacteria at start)
• Growth factor: \(\mathrm{b = 2}\) (since "doubles" means multiplies by 2)
• Time variable: \(\mathrm{t}\) = days after start of observation

Step 4: INFER the complete equation

• Substituting into \(\mathrm{y = a(b)^t}\):
• \(\mathrm{y = 44,000(2)^t}\)

Step 5: APPLY CONSTRAINTS to select the correct answer choice

• Looking at the options, this matches Choice D exactly

Answer: D. \(\mathrm{y = 44,000(2)^t}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse "doubles" with "half" and use 1/2 as the growth factor instead of 2.

They think: "If something doubles, maybe it's getting cut in half each time?" This fundamental misunderstanding of what "doubling" means in mathematical context leads them to write \(\mathrm{y = 44,000(1/2)^t}\).

This may lead them to select Choice C (\(\mathrm{y = 44,000(1/2)^t}\))

Second Most Common Error:

Poor INFER reasoning: Students recognize the exponential form but incorrectly position the initial value and growth factor, thinking the 2 should be the coefficient.

They reason: "The bacteria doubles, so I need a 2 in front, and 44,000 is what grows exponentially." This leads to the incorrect form \(\mathrm{y = 2(44,000)^t}\).

This may lead them to select Choice B (\(\mathrm{y = 2(44,000)^t}\))

The Bottom Line:

This problem tests whether students can correctly translate real-world exponential growth language into mathematical notation. The key insight is recognizing that "doubles" means the growth factor is 2, not that 2 should appear as a coefficient.