Explanation Given: The problem starts with an equilateral triangle at stage 0 and produces the next stage by replacing each edge with four edges of one-third the length and the process continues till infinity as shown in the Figure. Formula used: Limit rule states that lim x → ∞ a x = ∞ Calculation: (a) Since at the ( n − 1 ) t h stage, each stage of the polygon is replaced by 4 edges of 1 3 its length, therefore, the polygon at the n t h stage will be 4. 1 3 = 4 3 times the perimeter of the polygon at the previous or ( n − 1 ) t h stage. Let the perimeter of stage n and n − 1 is represented by P n and P n − 1 respectively. Then since the perimeter of stage n is 4 3 of the previous stage. It implies that P n = 4 3 P n − 1 Let n = 1 , P 1 = 4 3 P 0 Let n = 2 , P 2 = 4 3 P 1 = 4 3 ( 4 3 P 0 ) = ( 4 3 ) 2 P 0 Let n = 3 , P 3 = 4 3 P 2 = 4 3 [ ( 4 3 ) 2 P 0 ] ⇒ P n = ( 4 3 ) n P 0 As n is positive and 4 3 > 1 , therefore using the formula we get lim n → ∞ P n = ( 4 3 ) n P 0 (b) At the ( n − 1 ) t h stage, there are ( 3 ) 4 n − 1 edges in the snowflake...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 11.2, Problem 71E

To determine

(a) That the perimeter $P_{n}$ of the polygon at the $n^{t h}$ stage satisfies $P_{n} = \frac{4}{3} P_{n - 1}$ .We need to prove that $\lim_{n \to \infty} P_{n} = \infty$

(b) That the $(3) 4^{n - 1}$ new triangles are added at the $n^{t h}$ stage, each with area $\frac{A_{0}}{9^{n}} (for n \geq 1)$ . We need to prove that the total area of the snowflake is $\frac{8}{5} A_{0}$

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Chapter 11.2, Problem 70E

Chapter 11.3, Problem 1PQ

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Chapter 11 Solutions

CALCULUS (CLOTH)

Ch. 11.1 - Prob. 1PQ Ch. 11.1 - Prob. 2PQ Ch. 11.1 - Prob. 3PQ Ch. 11.1 - Prob. 4PQ Ch. 11.1 - Prob. 5PQ Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E

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Solution Summary: The author explains how the polygon at the n t h stage satisfies Pn = 4 3 P 1 and proves that the snowflake is 8 5 A 0.

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(a) That the perimeter Pn of the polygon at the nth stage satisfies Pn=43Pn−1.We need to prove that limn→∞Pn=∞ (b) That the (3)4n−1 new triangles are added at the nth stage, each with area A09n(for n≥1). We need to prove that the total area of the snowflake is 85A0

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Chapter 11 Solutions