https://file.aiquickdraw.com/mathbot/file/1725470373366e76ltna5.jpg

The problem in the image states:

"A sheet of paper of thickness 0.01 mm is repeatedly folded to fill up a gap of size 1.28 cm. The required number of folds is:

(A) 12
(B) 10
(C) 8
(D) 7"

### Solution:

We need to determine how many folds of the paper will be required to fill the gap of 1.28 cm, given that each sheet has a thickness of 0.01 mm.

1. **Initial Setup**:
   - Thickness of the paper = \(0.01 \text{ mm} = 0.001 \text{ cm}\)
   - The gap to be filled = \(1.28 \text{ cm}\)

2. **Folding Process**:
   Every time you fold the paper, the thickness doubles. So, after \( n \) folds, the thickness of the paper will be:
   \[
   \text{Thickness after } n \text{ folds} = 0.001 \times 2^n \text{ cm}
   \]

3. **Find \( n \)**:
   We need to find the smallest \( n \) such that:
   \[
   0.001 \times 2^n \geq 1.28
   \]
   This gives:
   \[
   2^n \geq \frac{1.28}{0.001} = 1280
   \]

4. **Logarithmic Calculation**:
   Taking the logarithm (base 2) on both sides:
   \[
   n \geq \log_2(1280)
   \]

   Let's calculate the value of \( n \).

I'll perform the calculation now.The required number of folds to fill the gap of 1.28 cm with a sheet of paper of thickness 0.01 mm is \( n = 11 \).

Since this is closest to the options, the correct answer would be none of the options listed, but based on general rounding rules, \( n \) could be approximated to the closest provided answer.

Would you like to explore further details, or have any questions?

### Related Questions:
1. How would the number of folds change if the paper's thickness was doubled?
2. What is the impact on the number of folds if the gap size is reduced by half?
3. How does the number of folds relate to the exponential growth of thickness?
4. Could the folding process eventually stop at a certain point due to practical limitations?
5. What would happen if the gap was measured in millimeters instead of centimeters?

**Tip:** In exponential growth scenarios like folding paper, the increase in thickness accelerates quickly after each fold, showing the power of exponential functions.

A sheet of paper of thickness 0.01 mm is repeatedly folded to fill up a gap of size 1.28 cm. The required number of folds is: (A) 12 (B) 10 (C) 8 (D) 7

Find out how many folds of a paper sheet of 0.01 mm thickness are required to fill a 1.28 cm gap. This problem involves exponential growth and logarithm calculations. Suitable for students in Grades 9-10.

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