The length of perimeter can represent a function what is [] because perimeter means the total length of the figure. The first perimeter of triangle is 3.
Let An= the area of the snowflake
When the n values are 0, 1, 2, and3, the length of the perimeter will be;
This is not constant to increase their area because the area of coming triangle will be decreasing according to increase the number of n. So, it needs to find how to change the area of coming triangle.
The area of triangles will be found as using the Pythagorean Theorem [ ].
Therefore the area of first triangle will be
= …①
The second triangle is able to find as calculate of the coming triangle’s area add to the first triangle which indicated ①. So, area of coming triangle will be
Area of one of the coming triangle is
A=
= …②
Thus the total area of second figure will be
+×３
= +
=
The coming triangle’s area has characteristic which is 1/9 of the first triangle.
②×９＝①
×９＝①
＝① [①＝]
Therefore, the coming triangle’s area will be 1/9 of the first triangle.
Moreover, the triangle can be separated into 9. The one of the small area of triangle will equal with coming triangle’s areas. That’s why the coming triangle’s area will be 1/9 of the first triangle area.
It is considerable that can say others area of coming triangle due to the same theory. So the function of will be [=]
= the area of coming triangle.
= the number of coming triangles.
= the area of figure which was found before of it.
- Using a GDC or a suitable graphing software package, create graphs of the four sets of values plotted against the value of n. Provide separate printed output for each graph.
The relationship between n and the number of side
The relationship between n and the number of a single side
The relationship between n and the length of perimeter
The relationship between n and area of snowflake
- For each of the graphs above, develop a statement in terms of n that generalizes the behavior shown in its graph. Explain how you arrived at your generalizations. Verify that your generalizations apply consistently to the sets of values produced in the table.
I got these functions below which have already indicated the reasons.
[= 4]
[ln =]
[]
[=]
These functions are fit with graphs which had already done no2.
[= 4]
Due to the n increase, the number of side also increases because each time the one side will be separated into 4.
Now we check that function is correct or not.
It has to be the same values in table below.
[= 4]
= 4×3 = 12
= 4×12=48
=4×48=192
Therefore, this function is correct.
[ln =]
Due to the n increase, the number of side decreases because a single side will be separated each time, so the length of a single side has to decrease.
Now, check the functions.
[ln =]
=
=
=
So, this function is correct either.
[]
Due to increase the value of n, the length of the perimeter also increases because of the number of sides are increasing.
Now, check the function
[]
1×3＝3
12×＝4
48×＝
192×=
Thus, this function is correct as well.
[=]
Due to value of n increase, the area also increases because it keeps expanding. The changing of area is going to a little because of the area of coming triangle is smaller and smaller.
Now, check the function.
[=]
== =
====
====
The function is correct as well.
-
Investigate what happens at n=4. Use your conjectures from step3 to obtains for,, and . Now draw a large diagram of one “side” (that is, one side of the original triangle that has been transformed) of the fractal at stage 4 and clearly verify your predictions.
[= 4]
= 4=4×192=768
[ln =]
===
[]
=768×=
[=]
====
Diagram of one of side when n=4.
-
Calculate values for,, and . You need not verify these answers.
[= 4]
= 4=4×()=12288
[ln =]
====
[]
=768×=
[=]
==
[====]
+=
-
Write down successive values of in term of. What pattern emerges?
[=] have been proved. However, the value ofcan’t get from this function because of anddoes not exist in this report. Thus find another characteristic as transform the function, the values of 1, 2, 3, and 4.
+
+
These function can be more simply thus
It is obvious that the function has characteristic thus it can define.
Let
Therefore the value of will be
(sum of )
- Explain what happens to the perimeter and area as n gets very large. What conclusion can you make about the area as n →∞? Comment on your results.
The perimeter and area will be infinite. By the solution of area of figure could represent [(sum of)]. That means when value of n is infinite, area also infinite as well. Therefore, the value of perimeter and area of snowflake are infinite when the n value is infinite either.
