A piece of paper is 0.01 cm thick. When you fold it once, it becomes 0.02 cm thick. If you fold it again, it doubles again to 0.04 cm thick. Each fold doubles the thickness of the paper. If there are 1000 meters in a kilometer and the Moon is 384, 000 kilometers away from the Earth, will the paper reach the Moon after 40 folds?
The Correct Answer and Explanation is :
To determine whether the paper would reach the Moon after 40 folds, we need to calculate the thickness of the paper after 40 folds and compare it to the distance from the Earth to the Moon.
Step 1: Understand the doubling process
The paper doubles in thickness with each fold. Starting with a thickness of 0.01 cm, after each fold, the thickness will double:
- After 1 fold: 0.01 cm × 2 = 0.02 cm
- After 2 folds: 0.02 cm × 2 = 0.04 cm
- And so on…
This means that the thickness of the paper after nn folds is given by: Thickness after n folds=0.01×2n cm\text{Thickness after } n \text{ folds} = 0.01 \times 2^n \text{ cm}
Step 2: Calculate the thickness after 40 folds
Using the formula, the thickness after 40 folds is: Thickness after 40 folds=0.01×240 cm\text{Thickness after 40 folds} = 0.01 \times 2^{40} \text{ cm}
Now, calculate 2402^{40}: 240=1,099,511,627,7762^{40} = 1,099,511,627,776
So, the thickness after 40 folds is: 0.01×1,099,511,627,776=10,995,116,277.76 cm0.01 \times 1,099,511,627,776 = 10,995,116,277.76 \text{ cm}
Step 3: Convert the thickness to kilometers
To convert from centimeters to kilometers, divide by 100,000 (since 100 cm = 1 meter, and 1,000 meters = 1 kilometer): Thickness in kilometers=10,995,116,277.76100,000=109,951.16 kilometers\text{Thickness in kilometers} = \frac{10,995,116,277.76}{100,000} = 109,951.16 \text{ kilometers}
Step 4: Compare the thickness to the distance to the Moon
The distance from the Earth to the Moon is 384,000 kilometers. After 40 folds, the paper’s thickness (109,951.16 kilometers) is far smaller than the distance to the Moon, which is 384,000 kilometers. Therefore, the paper will not reach the Moon after 40 folds. In fact, it falls short by about 274,048.84 kilometers.
Conclusion:
Even after 40 folds, the paper’s thickness (109,951.16 kilometers) is still much smaller than the distance to the Moon (384,000 kilometers). Therefore, the paper will not reach the Moon after 40 folds.