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Answer :
### Step-by-Step Solution
1. Understand the Problem:
- The thickness of a piece of paper is [tex]\(0.01 \, \text{cm}\)[/tex].
- The distance from the Earth to the Moon is [tex]\(384,000 \, \text{km}\)[/tex].
- We need to determine how many times we would need to fold the paper for its thickness to reach the Moon.
2. Convert the Distance to Consistent Units:
- Since the paper thickness is given in centimeters, convert the distance to the Moon from kilometers to centimeters.
- [tex]\(1 \, \text{km} = 100,000 \, \text{cm}\)[/tex]
- Therefore, [tex]\(384,000 \, \text{km} = 384,000 \times 100,000 = 38,400,000,000 \, \text{cm}\)[/tex].
3. Understanding the Folding Process:
- Each time you fold the paper, its thickness doubles.
- If [tex]\(n\)[/tex] is the number of folds, the thickness after [tex]\(n\)[/tex] folds will be [tex]\(0.01 \, \text{cm} \times 2^n\)[/tex].
4. Set Up the Equation:
- We need the thickness after [tex]\(n\)[/tex] folds to be at least equal to the distance to the Moon:
- [tex]\(0.01 \times 2^n \geq 38,400,000,000 \, \text{cm}\)[/tex]
5. Solve for [tex]\(n\)[/tex]:
- Isolate [tex]\(2^n\)[/tex]:
- [tex]\(2^n \geq \frac{38,400,000,000 \, \text{cm}}{0.01 \, \text{cm}}\)[/tex]
- [tex]\(2^n \geq 3,840,000,000,000\)[/tex]
6. Using Logarithms to Solve for [tex]\(n\)[/tex]:
- To solve [tex]\(2^n \geq 3,840,000,000,000\)[/tex], take the base-2 logarithm of both sides:
- [tex]\(n \geq \log_2(3,840,000,000,000)\)[/tex]
7. Calculate the Value:
- Using a calculator or logarithm table, you find:
- [tex]\( \log_2(3,840,000,000,000) \approx 41.84\)[/tex]
8. Determine the Number of Folds:
- Since [tex]\(n\)[/tex] must be an integer, round up [tex]\(41.84\)[/tex] to get the next whole number, which is [tex]\(42\)[/tex].
### Conclusion
You would need to fold the piece of paper 42 times for its thickness to reach the Moon.
1. Understand the Problem:
- The thickness of a piece of paper is [tex]\(0.01 \, \text{cm}\)[/tex].
- The distance from the Earth to the Moon is [tex]\(384,000 \, \text{km}\)[/tex].
- We need to determine how many times we would need to fold the paper for its thickness to reach the Moon.
2. Convert the Distance to Consistent Units:
- Since the paper thickness is given in centimeters, convert the distance to the Moon from kilometers to centimeters.
- [tex]\(1 \, \text{km} = 100,000 \, \text{cm}\)[/tex]
- Therefore, [tex]\(384,000 \, \text{km} = 384,000 \times 100,000 = 38,400,000,000 \, \text{cm}\)[/tex].
3. Understanding the Folding Process:
- Each time you fold the paper, its thickness doubles.
- If [tex]\(n\)[/tex] is the number of folds, the thickness after [tex]\(n\)[/tex] folds will be [tex]\(0.01 \, \text{cm} \times 2^n\)[/tex].
4. Set Up the Equation:
- We need the thickness after [tex]\(n\)[/tex] folds to be at least equal to the distance to the Moon:
- [tex]\(0.01 \times 2^n \geq 38,400,000,000 \, \text{cm}\)[/tex]
5. Solve for [tex]\(n\)[/tex]:
- Isolate [tex]\(2^n\)[/tex]:
- [tex]\(2^n \geq \frac{38,400,000,000 \, \text{cm}}{0.01 \, \text{cm}}\)[/tex]
- [tex]\(2^n \geq 3,840,000,000,000\)[/tex]
6. Using Logarithms to Solve for [tex]\(n\)[/tex]:
- To solve [tex]\(2^n \geq 3,840,000,000,000\)[/tex], take the base-2 logarithm of both sides:
- [tex]\(n \geq \log_2(3,840,000,000,000)\)[/tex]
7. Calculate the Value:
- Using a calculator or logarithm table, you find:
- [tex]\( \log_2(3,840,000,000,000) \approx 41.84\)[/tex]
8. Determine the Number of Folds:
- Since [tex]\(n\)[/tex] must be an integer, round up [tex]\(41.84\)[/tex] to get the next whole number, which is [tex]\(42\)[/tex].
### Conclusion
You would need to fold the piece of paper 42 times for its thickness to reach the Moon.
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