High School

Q11. The thickness of a piece of paper is 0.01 cm. The Moon is 384,000 km from the Earth. Assuming you can fold the piece of paper as many times as you like, how many times would you have to fold it for it to reach the Moon?

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.