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The population of a bacteria culture doubles every 2 minutes. Approximately how many minutes will it take for the population to grow from 1,000 to 500,000 bacteria?
Let's break down what this problem is asking us to do in plain language:
We start with 1,000 bacteria in our culture. Every 2 minutes, the population doubles - meaning it becomes twice as large as it was before. We want to know how much time passes before we reach 500,000 bacteria.
Think of it like this: if you had $100 and it doubled every hour, after 1 hour you'd have $200, after 2 hours you'd have $400, and so on. Similarly, our bacteria population keeps doubling at regular 2-minute intervals.
Process Skill: TRANSLATE - Converting the word "doubles every 2 minutes" into the mathematical understanding that population = previous population × 2 after each 2-minute period
Let's follow the bacteria population step by step, doubling it every 2 minutes:
Starting point (Time = 0 minutes): 1,000 bacteria
After 2 minutes: \(1{,}000 \times 2 = 2{,}000\) bacteria
After 4 minutes: \(2{,}000 \times 2 = 4{,}000\) bacteria
After 6 minutes: \(4{,}000 \times 2 = 8{,}000\) bacteria
After 8 minutes: \(8{,}000 \times 2 = 16{,}000\) bacteria
After 10 minutes: \(16{,}000 \times 2 = 32{,}000\) bacteria
After 12 minutes: \(32{,}000 \times 2 = 64{,}000\) bacteria
After 14 minutes: \(64{,}000 \times 2 = 128{,}000\) bacteria
After 16 minutes: \(128{,}000 \times 2 = 256{,}000\) bacteria
After 18 minutes: \(256{,}000 \times 2 = 512{,}000\) bacteria
Notice that after 16 minutes we have 256,000 bacteria (still below our target of 500,000), but after 18 minutes we have 512,000 bacteria (which exceeds our target).
Looking at our progression above, we can see that:
- At 16 minutes: 256,000 bacteria (still below 500,000)
- At 18 minutes: 512,000 bacteria (exceeds 500,000)
This means it takes exactly 9 doubling intervals to go from 1,000 to over 500,000 bacteria. Let's verify this count:
Interval 1: 1,000 → 2,000
Interval 2: 2,000 → 4,000
Interval 3: 4,000 → 8,000
Interval 4: 8,000 → 16,000
Interval 5: 16,000 → 32,000
Interval 6: 32,000 → 64,000
Interval 7: 64,000 → 128,000
Interval 8: 128,000 → 256,000
Interval 9: 256,000 → 512,000 ✓
Yes, we need 9 intervals to reach our target.
Since each doubling interval takes 2 minutes, and we need 9 intervals total:
Total time = Number of intervals × Time per interval
Total time = \(9 \times 2\) minutes = 18 minutes
It takes 18 minutes for the bacteria population to grow from 1,000 to 500,000 (actually reaching 512,000 at the 18-minute mark).
Looking at our answer choices: (A) 10, (B) 12, (C) 14, (D) 16, (E) 18
The answer is (E) 18.
1. Misunderstanding what "doubles every 2 minutes" means
Students might think this means the population increases by 2 every 2 minutes, or that it takes 2 minutes total to double, rather than understanding that every 2-minute period results in the population becoming twice its previous size.
2. Confusion about when to stop counting
Students may think they need to find the exact time when the population equals 500,000, not realizing that since doubling happens at discrete intervals, they need to find when it first exceeds or reaches approximately 500,000.
3. Setting up the wrong mathematical relationship
Instead of recognizing this as an exponential growth pattern (\(\mathrm{Population} = \mathrm{Initial} \times 2^n\)), students might try to set up a linear relationship or use incorrect formulas.
1. Arithmetic errors in the doubling sequence
When manually calculating the doubling pattern (1,000 → 2,000 → 4,000 → 8,000...), students often make multiplication errors, especially with larger numbers like \(128{,}000 \times 2\) or \(256{,}000 \times 2\).
2. Losing track of time intervals
Students may correctly double the population but incorrectly count the number of intervals or forget that each interval represents 2 minutes, leading to errors like saying 9 intervals = 9 minutes instead of 18 minutes.
3. Miscounting the number of doublings needed
Students might stop one interval too early (at 256,000 after 8 intervals) thinking it's "close enough" to 500,000, or miscount the total number of doublings required.
1. Converting intervals to time incorrectly
Even after correctly determining that 9 intervals are needed, students might forget to multiply by 2 minutes per interval, selecting 9 or some other incorrect time value instead of 18 minutes.
2. Selecting the time for the previous interval
Students might correctly calculate that after 16 minutes there are 256,000 bacteria, but incorrectly conclude that 16 minutes is "close enough" and select answer choice (D) 16 instead of (E) 18.