Stage Area of Rectangle (in2) 0 187/2 1 187/4 2 187/8 3 187/16 4 187/32 5 187/64 6 187/128 2.
3. The limit seems to be zero because, with an increase in the stage number, the y-values get closer and closer to zero.
4. Pattern to find the area of any stage (e.g. 7, 8, 9, and 100 etc.): A = 187/ 2(n+1); where n = the stage number and A = area in inches squared. -To find the area of stage 100, one must simply replace n with 100 in the aforementioned equation. A = 187/ 2(100+1) = 187/2101 in2
Stage Area of Rectangle (in2) 7 187/256 8 187/512 9 187/1024 100 187/2101
6. The limit seems to be zero, because with an increase in the stage number, the y-values get closer and closer to zero.
7. Pattern to find the perimeter of any stage (except 0 and 1): - To find the perimeter of any odd numbered stage (e.g. 7, 9, and 11 etc.): P = 28/2 (v/2); where P = the perimeter in inches, v = n-1, n = the stage number, - To find the perimeter of any even numbered stage (e.g. 8, 10, and 12 etc.): P = 39/2(n/2); where P = the perimeter in inches, and n = the stage number. - To find the perimeter of stage 100 (knowing “100” is an even number) replace n with 100 in the aforementioned ed equation for even numbered stages: P = 39/2(100/2) = 39/250
9. All the triangles in the Sierpinski triangle are similar to each, from the smallest to the largest. For example, the inscribed and shaded triangle in stage 1 is similar to the larger un-shaded triangle in stage 0.
There is no limit because the y-values constantly increase with an increase in the stage number.
12. Pattern to find the perimeter of any stage (e.g. 7,8,9, and 100): P = 1.5n (48); Where P = the perimeter and n = the stage. - To find the perimeter of stage 100, simply replace n with 100 in the above equation.
Stage Total Area (square units) 0 111 1 83.25 2 66.6 3 55.5 4 333/7
13.
14. The limit is zero because, with an increase in the stage number, the y-values get closer and closer to zero.
15. Pattern to find the area of any stage (e.g. 7,8,9, and 100 etc.): A = 333/(n+3); where A = the area in square units and n = the stage number. -To find the area of stage 100, simply replace n with 100 in the equation above regarding area. A = 333/(100+3) = 333/103
17. Koch’s snowflake does show self-similarity because segments/triangles/shapes added at each stage resemble the segments/triangles/shapes that were added in the previous stages.
Stage Perimeter 0 81 1 108 2 144 3 180 4 186 18.
19.
The Sequence of perimeters does not have a limit because the y-values Continue to increase with every stage.
20. Pattern to find the total perimeter of any stage (e.g. 7, 8, 9, and 100):
- To find the total perimeter of stage 100, replace n with 100 in the equation above ( )
Do it Again, Sam.
ReplyDeletePaper Rectangles
1.
Stage Area of Rectangle (in2)
0 187/2
1 187/4
2 187/8
3 187/16
4 187/32
5 187/64
6 187/128
2.
3.
The limit seems to be zero because, with an increase in the stage
number, the y-values get closer and closer to zero.
4. Pattern to find the area of any stage (e.g. 7, 8, 9, and 100 etc.):
A = 187/ 2(n+1); where n = the stage number and A = area in inches squared.
-To find the area of stage 100, one must simply replace n with 100 in the aforementioned equation.
A = 187/ 2(100+1) = 187/2101 in2
Stage Area of Rectangle (in2)
7 187/256
8 187/512
9 187/1024
100 187/2101
Stage Perimeter of Rectangle (in)
0 39
1 28
2 19.5
3 14
4 9.75
5 7
6 4.875
5.
6.
The limit seems to be zero, because with an increase in the stage
number, the y-values get closer and closer to zero.
7. Pattern to find the perimeter of any stage (except 0 and 1):
- To find the perimeter of any odd numbered stage (e.g. 7, 9, and 11 etc.):
P = 28/2 (v/2); where P = the perimeter in inches, v = n-1, n = the stage number,
- To find the perimeter of any even numbered stage (e.g. 8, 10, and 12 etc.):
P = 39/2(n/2); where P = the perimeter in inches, and n = the stage number.
- To find the perimeter of stage 100 (knowing “100” is an even number) replace n with 100 in the
aforementioned ed equation for even numbered stages:
P = 39/2(100/2) = 39/250
Stage Perimeter of rectangle (in)
7 3.5
8 2.4375
9 1.75
100 39/250
The Sierpinski Triangle
Stage Side Perimeter Area
0 16 48 111
1 8 72 83.25
2 4 108 66.6
3 2 162 55.5
4 1 243 333/7
8.
9. All the triangles in the Sierpinski triangle are similar to each, from the smallest to the largest. For example, the inscribed and shaded triangle in stage 1 is similar to the larger un-shaded triangle in stage 0.
Stage Perimeter of Triangles
0 48
1 72
2 108
3 162
4 243
10.
11.
There is no limit because the y-values constantly increase
with an increase in the stage number.
12. Pattern to find the perimeter of any stage (e.g. 7,8,9, and 100):
P = 1.5n (48); Where P = the perimeter and n = the stage.
- To find the perimeter of stage 100, simply replace n with 100 in
the above equation.
Stage Perimeter
7 820.125
8 1230.1875
9 1845.28125
100 1.5100 (48)
Stage Total Area (square units)
0 111
1 83.25
2 66.6
3 55.5
4 333/7
13.
14.
The limit is zero because, with an increase in the stage number,
the y-values get closer and closer to zero.
15. Pattern to find the area of any stage (e.g. 7,8,9, and 100 etc.):
A = 333/(n+3); where A = the area in square units and n = the stage number.
-To find the area of stage 100, simply replace n with 100 in the equation above regarding area.
A = 333/(100+3) = 333/103
Stage Area (square units)
7 33.3
8 333/11
9 27.75
100 333/103
16.
17. Koch’s snowflake does show self-similarity because segments/triangles/shapes added at each stage resemble the segments/triangles/shapes that were added in the previous stages.
Stage Perimeter
0 81
1 108
2 144
3 180
4 186
18.
19.
The Sequence of perimeters does not have a limit because the y-values
Continue to increase with every stage.
20. Pattern to find the total perimeter of any stage (e.g. 7, 8, 9, and 100):
- To find the total perimeter of stage 100, replace n with 100 in the equation above ( )
Stage Total Perimeter
7
8
9
100
Stage Area (units square)
0 316
1 437.5
2 491.5
3 509.5
4 527.5
21.
22. There is no limit because the y-values (the area) increase with an
increase in the stage number (x-values).
23. Pattern to find the area of any stage (e.g. 7, 8, 9, and 100):
-To find the area of stage 100, replace n with 100 in the above equation ( ).
Stage Area (square units)
7
8
9
100