A square has a side of 2 dm. A new square is formed by joining the midpoints of the sides. If this procedure is repeated 6 times, what is the perimeter of the smallest square?
After joining the midpoints of the sides of a square, a smaller square is formed inside it. The ratio of the side length of the smaller square to the side length of the original square is $1:2$. Thus, the side length of each square after the first square is half the side length of the previous square in the sequence.
Therefore, the sequence of side lengths of the squares is $2, 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}$ dm. The perimeter of each square is four times its side length, so the sequence of perimeters is $8, 4, 2, 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}$ dm.
The perimeter of the smallest square is $\boxed{\frac{1}{8}}$ dm.
To solve this problem, we need to calculate the perimeter of each square as the procedure is repeated 6 times.
1. The initial square has a side length of 2 dm. The perimeter of the first square is equal to 4 times the side length.
Perimeter of first square = 4 * 2 dm = 8 dm
2. The second square is formed by joining the midpoints of the sides of the first square. The side length of the second square is half that of the first square.
Side length of second square = 2 dm / 2 = 1 dm
Perimeter of second square = 4 * 1 dm = 4 dm
3. The third square is formed by joining the midpoints of the sides of the second square. The side length of the third square is half that of the second square.
Side length of third square = 1 dm / 2 = 0.5 dm
Perimeter of third square = 4 * 0.5 dm = 2 dm
4. Continuing this process, we find that the side length of the fourth square is 0.25 dm, the side length of the fifth square is 0.125 dm, and the side length of the sixth square is 0.0625 dm.
5. Now, let's calculate the perimeter of the sixth (smallest) square.
Perimeter of sixth square = 4 * 0.0625 dm = 0.25 dm
Therefore, the perimeter of the smallest square, after the procedure is repeated 6 times, is 0.25 dm.
To find the perimeter of the smallest square after the procedure is repeated 6 times, we need to understand the pattern and process involved.
Let's start with the first square with a side of 2 dm. The midpoints of the sides are joined to form a new square inside the original square. The side length of this new square is half of the original square's side length, which is 1 dm.
The process is repeated 6 times. Each time, a new square is formed inside the previous square by joining the midpoints of its sides. The side length of each new square is half of the previous square's side length. Let's see the pattern:
1st square: side length = 2 dm
2nd square: side length = 1 dm (half of the previous square)
3rd square: side length = 0.5 dm (half of the previous square)
4th square: side length = 0.25 dm (half of the previous square)
5th square: side length = 0.125 dm (half of the previous square)
6th square: side length = 0.0625 dm (half of the previous square)
Now let's calculate the perimeter of the smallest square formed after 6 iterations.
The perimeter of a square is calculated by multiplying the side length by 4:
Perimeter of 6th square = side length * 4
Perimeter of 6th square = 0.0625 dm * 4
Perimeter of 6th square = 0.25 dm
Therefore, the perimeter of the smallest square after 6 iterations is 0.25 dm.