In this exercise, we are finding the perimeter of a Sierpinski gasket. (By "perimeter," we mean the total distance around all of the filled in regions, not the distance around the outside only.) To create the Sierpinski gasket, use the following steps. 1. Draw an equilateral triangle and fill in its interior. 2. Put a hole in the triangle: Find the midpoints of the sides of the triangle. Connect the midpoints and form a new triangle. Form a hole by removing the new triangle. 3. The original triangle now has a triangular hole in its center, surrounded by three other triangles. Subdivide each of these three triangles into four smaller triangles. Remove each of the central triangles. 4. Continue this process indefinitely. (a) Use your drawings of the Sierpinski gasket to explain each of the calculations in the chart shown below, where "triangle" refers to a filled-in triangle, not a triangular hole. In step 1, there is only one triangle. If each side is 1 ft. in length, then the perimeter is ft. In step 2, the original triangle is modified by removing a center triangle. This results in smaller triangles. Each side of the original triangle now has two triangles with sides of length ft. Thus, the perimeter of one of these smaller triangles is ft. Since there are three such triangles, the total perimeter is ft. Step Number of Triangles Length of Each Side Perimeter of One Triangle Total Perimeter of All Triangles 1 1 1 ft 3 · 1 ft = 3 ft 1 · 3 ft = 3 ft 2 3* 1 /2 · 1 ft = 1/2 ft 3 · 1/2 ft = 3/2 ft 3 · 3/2 ft = 9/2 ft 3 4 5 6 *3 not 4, because only three triangles are filled in. (b) Use your drawings of the Sierpinski gasket to complete the chart. Step Number of Triangles Length of Each Side Perimeter of One Triangle Total Perimeter of All Triangles 1 1 1 ft 3 · 1 ft = 3 ft 1 · 3 ft = 3 ft 2 3* 1 /2 · 1 ft = 1/2 ft 3 · 1/2 ft = 3/2 ft 3 · 3/2 ft = 9/2 ft 3 4 5 6 *3 not 4, because only three triangles are filled in. (c) By what factor does the number of triangles increase? (d) By what factor does the length of each side decrease? (e) By what factor does the perimeter of one triangle change? Is the perimeter of one triangle increasing or decreasing? increasingdecreasing (f) By what factor does the total perimeter of all triangles change? Is the total perimeter of all triangles increasing or decreasing? increasingdecreasing (g) Use your answers to parts (a)-(f) to find a formula for the total perimeter P of all triangles at step n. Pn = (h) Use your answer to part (g) and your calculator to find the perimeter of the Sierpinski gasket. (By "perimeter," we mean the total distance around all of the filled in regions, not the distance around the outside only. Enter INFINITY for ∞ if needed.) P = ft

In this exercise, we are finding the perimeter of a Sierpinski gasket. (By "perimeter," we mean the total distance around all of the filled in regions, not the distance around the outside only.) To create the Sierpinski gasket, use the following steps. 1. Draw an equilateral triangle and fill in its interior. 2. Put a hole in the triangle: Find the midpoints of the sides of the triangle. Connect the midpoints and form a new triangle. Form a hole by removing the new triangle. 3. The original triangle now has a triangular hole in its center, surrounded by three other triangles. Subdivide each of these three triangles into four smaller triangles. Remove each of the central triangles. 4. Continue this process indefinitely. (a) Use your drawings of the Sierpinski gasket to explain each of the calculations in the chart shown below, where "triangle" refers to a filled-in triangle, not a triangular hole. In step 1, there is only one triangle. If each side is 1 ft. in length, then the perimeter is ft. In step 2, the original triangle is modified by removing a center triangle. This results in smaller triangles. Each side of the original triangle now has two triangles with sides of length ft. Thus, the perimeter of one of these smaller triangles is ft. Since there are three such triangles, the total perimeter is ft. Step Number of Triangles Length of Each Side Perimeter of One Triangle Total Perimeter of All Triangles 1 1 1 ft 3 · 1 ft = 3 ft 1 · 3 ft = 3 ft 2 3* 1 /2 · 1 ft = 1/2 ft 3 · 1/2 ft = 3/2 ft 3 · 3/2 ft = 9/2 ft 3 4 5 6 3 not 4, because only three triangles are filled in. (b) Use your drawings of the Sierpinski gasket to complete the chart. Step Number of Triangles Length of Each Side Perimeter of One Triangle Total Perimeter of All Triangles 1 1 1 ft 3 · 1 ft = 3 ft 1 · 3 ft = 3 ft 2 3 1 /2 · 1 ft = 1/2 ft 3 · 1/2 ft = 3/2 ft 3 · 3/2 ft = 9/2 ft 3 4 5 6 *3 not 4, because only three triangles are filled in. (c) By what factor does the number of triangles increase? (d) By what factor does the length of each side decrease? (e) By what factor does the perimeter of one triangle change? Is the perimeter of one triangle increasing or decreasing? increasingdecreasing (f) By what factor does the total perimeter of all triangles change? Is the total perimeter of all triangles increasing or decreasing? increasingdecreasing (g) Use your answers to parts (a)-(f) to find a formula for the total perimeter P of all triangles at step n. Pn = (h) Use your answer to part (g) and your calculator to find the perimeter of the Sierpinski gasket. (By "perimeter," we mean the total distance around all of the filled in regions, not the distance around the outside only. Enter INFINITY for ∞ if needed.) P = ft

Name: In this exercise, we are finding the perimeter of a Sierpinski gasket.
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Description: In this exercise, we are finding the perimeter of a Sierpinski gasket. (By "perimeter," we mean the total distance around all of the filled in regions, not the distance around the outside only.) To create the Sierpinski gasket, use the following step

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

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Problem 40E: As in Exercise 39, find the volume of the box if four congruent squares with sides of length 6 in....

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1.	Draw an equilateral triangle and fill in its interior.
2.	Put a hole in the triangle: Find the midpoints of the sides of the triangle. Connect the midpoints and form a new triangle. Form a hole by removing the new triangle.
3.	The original triangle now has a triangular hole in its center, surrounded by three other triangles. Subdivide each of these three triangles into four smaller triangles. Remove each of the central triangles.
4.	Continue this process indefinitely.

(a) Use your drawings of the Sierpinski gasket to explain each of the calculations in the chart shown below, where "triangle" refers to a filled-in triangle, not a triangular hole.

In step 1, there is only one triangle. If each side is 1 ft. in length, then the perimeter is ft. In step 2, the original triangle is modified by removing a center triangle. This results in smaller triangles. Each side of the original triangle now has two triangles with sides of length ft. Thus, the perimeter of one of these smaller triangles is ft. Since there are three such triangles, the total perimeter is ft.

Step	Number of Triangles	Length of Each Side	Perimeter of One Triangle	Total Perimeter of All Triangles
1	1	1 ft	3 · 1 ft = 3 ft	1 · 3 ft = 3 ft
2	3*	1 /2 · 1 ft = 1/2 ft	3 · 1/2 ft = 3/2 ft	3 · 3/2 ft = 9/2 ft
3
4
5
6

*3 not 4, because only three triangles are filled in.

(b) Use your drawings of the Sierpinski gasket to complete the chart.

Step	Number of Triangles	Length of Each Side	Perimeter of One Triangle	Total Perimeter of All Triangles
1	1	1 ft	3 · 1 ft = 3 ft	1 · 3 ft = 3 ft
2	3*	1 /2 · 1 ft = 1/2 ft	3 · 1/2 ft = 3/2 ft	3 · 3/2 ft = 9/2 ft
3
4
5
6

*3 not 4, because only three triangles are filled in.

(c) By what factor does the number of triangles increase?

(d) By what factor does the length of each side decrease?

(e) By what factor does the perimeter of one triangle change?

Is the perimeter of one triangle increasing or decreasing?

increasingdecreasing

(f) By what factor does the total perimeter of all triangles change?

Is the total perimeter of all triangles increasing or decreasing?

increasingdecreasing

(g) Use your answers to parts (a)-(f) to find a formula for the total perimeter P of all triangles at step n.
P_n =

(h) Use your answer to part (g) and your calculator to find the perimeter of the Sierpinski gasket. (By "perimeter," we mean the total distance around all of the filled in regions, not the distance around the outside only. Enter INFINITY for ∞ if needed.)
P = ft

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