- Translate the problem requirements: We need to find what the scaled distance from Earth to the nearest star would be when the Earth-moon distance (\(380{,}000\) km) is represented by 1 cm. The actual star distance is \(4 \times 10^{13}\) km.
- Set up the scaling ratio: Determine the scale factor by comparing the actual Earth-moon distance to its scaled representation.
- Apply the scale to the star distance: Use the same scaling ratio to convert the actual Earth-star distance to its scaled equivalent.
- Compare with answer choices: Since answer choices are orders of magnitude apart, we can use approximation to identify the closest match.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we're given and what we need to find:
What we know:
• Distance from Earth to nearest star =
\(4 \times 10^{13}\) km
• Distance from Earth to moon =
\(380{,}000\) km
What we need to find:
If we shrink everything down so that the Earth-moon distance becomes just 1 centimeter, what would the Earth-star distance become?
Think of it like creating a scale model - imagine you're building a model where you represent the Earth-moon distance with a 1 cm stick. How long would you need to make the stick representing the Earth-star distance?
Process Skill: TRANSLATE - Converting the scaling scenario into a mathematical relationship
2. Set up the scaling ratio
To create our scale model, we need to figure out our "shrinking factor."
In simple terms: we're taking
\(380{,}000\) km and making it equal to 1 cm.
This means our scaling ratio is:
\(380{,}000\) km in real life = 1 cm in our model
So our shrinking factor is
\(380{,}000\) km per 1 cm, or we can think of it as:
Every
\(380{,}000\) km in real life becomes 1 cm in our model
Mathematically: Scale factor =
\(380{,}000\) km/cm
3. Apply the scale to the star distance
Now we use the same shrinking factor on the star distance.
The star is
\(4 \times 10^{13}\) km away. Using our scale:
Scaled star distance = (Actual star distance) ÷ (Scale factor)
Scaled star distance =
\((4 \times 10^{13} \text{ km}) \div (380{,}000 \text{ km/cm})\)
Let's work with the numbers step by step:
•
\(4 \times 10^{13} = 40{,}000{,}000{,}000{,}000\) km
•
\(380{,}000 = 3.8 \times 10^5\) km
So: Scaled distance =
\((4 \times 10^{13}) \div (3.8 \times 10^5)\)
=
\(\frac{4}{3.8} \times \frac{10^{13}}{10^5}\)
=
\(\frac{4}{3.8} \times 10^8\)
Since
\(\frac{4}{3.8}\) is approximately 1.05 (very close to 1):
Scaled distance ≈
\(1 \times 10^8\) cm =
\(100{,}000{,}000\) cm
4. Compare with answer choices
We found the scaled distance is about \(100{,}000{,}000\) cm, but the answer choices are in kilometers!
Let's convert:
\(100{,}000{,}000\) cm =
\(1{,}000{,}000\) m =
\(1{,}000\) km
Looking at our answer choices:
- 10 km
- 100 km
- 1,000 km ← This matches our result!
- 100,000 km
- 1,000,000 km
Final Answer
The scaled down distance from Earth to the nearest star outside our solar system would be closest to 1,000 kilometers.
Answer: C
Common Faltering Points
Errors while devising the approach
1. Misunderstanding the scaling concept:
Students often confuse which distance should be the reference point. They might try to scale the star distance to 1 cm instead of the moon distance, completely reversing the scaling relationship. The question clearly states "the distance from Earth to our moon was represented by 1 centimeter" - this is the fixed reference point for scaling.
2. Setting up the ratio incorrectly:
Many students struggle with whether to multiply or divide by the scale factor. They might set up the ratio as (moon distance) ÷ (star distance) instead of understanding that they need to find how many "moon distances" fit into the "star distance" to apply the same scaling factor.
Errors while executing the approach
1. Scientific notation calculation errors:
When calculating
\((4 \times 10^{13}) \div (3.8 \times 10^5)\), students frequently make mistakes with exponent subtraction, getting
\(10^7\) instead of
\(10^8\), or miscalculating
\(\frac{4}{3.8}\), leading to significantly different final answers.
2. Unit conversion mistakes during calculation:
Students might forget to properly handle the centimeter units throughout the calculation, or accidentally convert units too early in the process, leading to confusion about whether their intermediate results are in centimeters, meters, or kilometers.
Errors while selecting the answer
1. Final unit conversion oversight:
After correctly calculating
\(100{,}000{,}000\) cm, students often forget that the answer choices are in kilometers, not centimeters. They might select choice E (1,000,000) thinking their answer of
\(100{,}000{,}000\) is closest to this choice, not realizing they need to convert cm to km first.
2. Magnitude confusion:
Students may get the right order of magnitude in centimeters but then incorrectly convert to kilometers, confusing whether
\(100{,}000{,}000\) cm equals 100 km, 1,000 km, or 10,000 km due to rushing through the conversion or misremembering the conversion factors.
