- Translate the problem requirements: We need to find the population 12 hours ago, given that 4 hours from now the population will be \(1.28 \times 10^6\), and the population doubles every 4 hours. This means we're looking 16 hours backward total from the future reference point.
- Establish the timeline and reference points: Set up a clear timeline showing the relationship between different time points and use the future population as our starting reference.
- Apply the doubling pattern in reverse: Since population doubles every 4 hours going forward, it halves every 4 hours going backward. Calculate step-by-step moving backward through each 4-hour interval.
- Verify the answer against choices: Check our calculated result against the given options to ensure we have the correct order of magnitude and exact value.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know and what we need to find:
What we know:
- Four hours from now: population = \(1.28 \times 10^6\)
- Population doubles every 4 hours
What we need to find: The population 12 hours ago
Think of it this way: if we're standing at "now" and looking 4 hours into the future, then looking 12 hours into the past, we're actually looking at a total time difference of 4 + 12 = 16 hours between these two time points.
Process Skill: TRANSLATE - Converting the time relationships into a clear mathematical framework
2. Establish the timeline and reference points
Let's create a simple timeline to visualize the problem:
Timeline (moving left to right):
- 12 hours ago → 8 hours ago → 4 hours ago → NOW → 4 hours from now
- ? → ? → ? → ? → \(1.28 \times 10^6\)
Since the population doubles every 4 hours, we can see there are exactly 4 intervals of 4 hours each between "12 hours ago" and "4 hours from now".
This means the population will double 4 times as we move from "12 hours ago" to "4 hours from now".
3. Apply the doubling pattern in reverse
Here's the key insight: if population doubles every 4 hours going forward, then it halves every 4 hours going backward.
Starting from 4 hours in the future (\(1.28 \times 10^6\)) and working backward:
Step by step calculation:
- 4 hours from now: \(1.28 \times 10^6\)
- NOW (go back 4 hours): \((1.28 \times 10^6) \div 2 = 0.64 \times 10^6 = 6.4 \times 10^5\)
- 4 hours ago: \((6.4 \times 10^5) \div 2 = 3.2 \times 10^5\)
- 8 hours ago: \((3.2 \times 10^5) \div 2 = 1.6 \times 10^5\)
- 12 hours ago: \((1.6 \times 10^5) \div 2 = 0.8 \times 10^5 = 8.0 \times 10^4\)
Alternatively, we can think of this as: the population 12 hours ago was the current future population divided by \(2^4\) (since it doubles 4 times):
Population 12 hours ago = \((1.28 \times 10^6) \div 16 = 0.08 \times 10^6 = 8.0 \times 10^4\)
4. Verify the answer against choices
Our calculated answer is \(8.0 \times 10^4\).
Looking at the answer choices:
- \(6.4 \times 10^2\) (much too small)
- \(8.0 \times 10^4\) (matches our answer!)
- \(1.6 \times 10^5\) (this would be 8 hours ago)
- \(3.2 \times 10^5\) (this would be 4 hours ago)
- \(8.0 \times 10^6\) (much too large)
Verification check: If the population was
\(8.0 \times 10^4\) twelve hours ago, then:
- 8 hours ago: \(8.0 \times 10^4 \times 2 = 1.6 \times 10^5\)
- 4 hours ago: \(1.6 \times 10^5 \times 2 = 3.2 \times 10^5\)
- Now: \(3.2 \times 10^5 \times 2 = 6.4 \times 10^5\)
- 4 hours from now: \(6.4 \times 10^5 \times 2 = 1.28 \times 10^6\) ✓
Final Answer
The population 12 hours ago was \(8.0 \times 10^4\).
The correct answer is B. \(8.0 \times 10^4\).
Common Faltering Points
Errors while devising the approach
- Misunderstanding the timeline reference points: Students often get confused about which time point to use as their starting reference. They might try to work forward from "now" instead of backward from "4 hours from now," or they might incorrectly calculate the total time span. The key insight that there are exactly 16 hours (4 + 12) between the two target time points is frequently missed.
- Confusing the direction of the doubling pattern: Students may struggle with the concept that if population doubles going forward in time, it must be halved when going backward in time. They might attempt to double the population when moving backward, which would give a completely incorrect answer.
- Miscounting the number of doubling intervals: Students often fail to carefully count that there are exactly 4 intervals of 4 hours each between "12 hours ago" and "4 hours from now." They might count 3 intervals or 5 intervals, leading to incorrect exponential calculations.
Errors while executing the approach
- Arithmetic errors with scientific notation: When performing calculations like \((1.28 \times 10^6) \div 2 = 0.64 \times 10^6\), students frequently make errors in handling the decimal places or converting between different forms of scientific notation (e.g., \(0.64 \times 10^6 = 6.4 \times 10^5\)).
- Incorrect exponent calculations: When using the shortcut method of dividing by \(2^4 = 16\), students might miscalculate the exponent (using \(2^3 = 8\) or \(2^5 = 32\) instead) or make errors in the division: \((1.28 \times 10^6) \div 16 = 0.08 \times 10^6\).
- Sequential division errors: In the step-by-step backward calculation, students might lose track of their intermediate results or make computational mistakes when repeatedly dividing by 2, especially when converting scientific notation at each step.
Errors while selecting the answer
- Selecting intermediate calculation results: Students might accidentally select answer choices that correspond to intermediate time points in their calculation. For example, choosing C (\(1.6 \times 10^5\)) which represents 8 hours ago, or D (\(3.2 \times 10^5\)) which represents 4 hours ago, instead of the correct answer for 12 hours ago.
- Failing to verify the final answer: Students may not perform the forward verification check to confirm their answer. If they had calculated incorrectly and selected a wrong choice, working forward from their selected answer to see if it reaches \(1.28 \times 10^6\) after 16 hours would reveal the error.
