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Obtain a square piece of paper and measure its sides. Can you predict what the perimeter will be if the paper is folded in half? Test your prediction. Can you predict a new perimeter if the paper is folded again in the other direction? It is said that no piece of paper can be folded more than seven times in half. If it were possible to fold a piece 100 times what would the perimeter and area of your paper be?
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A square piece of paper with length of side L has an area L^2 and perimeter 4*L
If the sheet of paper is folded in half, the area is `L^2/2` and the perimeter is 2*L + 2*(L/2) = 2*L + L = 3L
If the sheet of paper is folded again, this time in the other direction, the area is `L^2/4` and the perimeter is 2*(L/2) + L = 2L
As the sheet is folded, the area becomes half each time and the perimeter is `4*sqrt(area)`
If it were sheet were folded 100 times, the final area would be `L^2/2^100` where `L^2` is the initial area. The final perimeter would be `4*sqrt(L^2/2^100)` = `4*L*(1/2^50)`
If the initial length of the sides is L, after the sheet is folded 100 times, the final area is `L^2/2^100` and the final perimeter is `4*L*(1/2^50)`
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