Hello,
Here is a question with something i've never heard of.
These are the first two stages of a fractal known as the Sierpinski carpet. The carpet begins with a square (stage 1). The square is cut into 9 congruent squares and the middle square is removed (stage 2).
Draw stage 3 using the method described to get from stage 1 to stage 2.
To the nearest whole percent, what percent of the original square remains in stage 3? Show your work and explain your answer.
ok, I realize that I should cut each remaining square into 9 quadrants leaving the middle open of each of them.
Now, I just don't know how to get the percent of the original square that remains in stage 3.
In stage one say the square is 9 units by 9 units thus 81 square units in area.
Then we cut it into 9 units each 3 units by three units so each 9 square units in area. We remove one so we only have 8 left each 9 square units in area.
Now for stage 3 we split each of those 8 squares into 9 squares each one unit by one unit or one unit square. We now divide one of those 1 by 1 squares into 9 squares each 1/3 by 1/3 so we can remove the middle one. The area left is 1 minus the middle which is 1/3*1/3 = 1/9 so it is 8/9
so we started with a square of 9 by 9 or 81 square units and now we have a little square with area of 8/9 square units so
(8/9)/81 is the ratio of area of one of the final little carpets to the original carpet.
Remember you have 8 of those little carpets.
This might help, nice drawing of it.
(Broken Link Removed)
To draw stage 3 of the Sierpinski carpet, we will apply the same process that was used to transition from stage 1 to stage 2. Let's break down the steps:
1. Start with a square in stage 2.
2. Divide the square into 9 congruent smaller squares, just like in stage 2.
3. Remove the middle square, leaving us with 8 smaller squares.
4. Repeat this process for each of the remaining 8 squares, subdividing them into 9 smaller squares and removing the middle square each time.
Continuing this process, we can generate subsequent stages of the Sierpinski carpet. Let's calculate the percentage of the original square that remains in stage 3.
In stage 1, we have a single square, which is 100% of the original square.
In stage 2, we divide the square into 9 congruent smaller squares. Since we remove the middle square, we are left with 8 smaller squares. Each of these smaller squares is one-ninth the size of the original square. Therefore, the total area of the 8 smaller squares is (8/9) of the original square.
Now, let's move to stage 3 by applying the same process again. Each of the 8 smaller squares from stage 2 will be divided into 9 smaller squares, and the middle square will be removed.
Since each of the 8 smaller squares is one-ninth the size of the original square, we'll have 8 * 8 = 64 smaller squares in total after the third stage.
Therefore, the remaining area of the original square in stage 3 is (64/81) of the original square. To find the percentage, we can calculate:
(64/81) * 100 = 79.01%
So, to the nearest whole percent, approximately 79% of the original square remains in stage 3 of the Sierpinski carpet.
Remember, the key to finding the solution is understanding the pattern and applying it recursively to find subsequent stages.