Reaching for Burj Khalifa
Burj Khalifa is currently the highest building in the world with the height of 829.8 meters, and you're thinking of a very odd way to reach its top.
That is, you want to fold a gigantic piece of paper of 0.1 millimeter thickness multiple times to increase the height of the folded paper vertically. Every time you fold the paper in halves, the total paper thickness doubles, and the paper's area is so infinite that you can continue folding as many times as you'd like.
At least how many times will you need to fold such paper so that it surpasses the top of the building?
The answer is 23.
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When we fold the paper for n times, the total height(thickness) = 0 . 0 0 0 1 × ( 2 n ) meters.
In order to reach the top of the building, we can equalize the building's height with the exponent:
0 . 0 0 0 1 × ( 2 n ) meters = 829.8 meters
2 n = 8298000
n = l o g 2 l o g 8 2 9 8 0 0 0 ≈22.98
Therefore, to surpass the top, we need to fold the paper at least 23 times.