Explanation Given : A unit ball in R n , n=8,9,10 Formula Used : A 2 m = π m m ! A 2 m + 1 = 2 m + 1 π m 1 × 3 × 5 × ⋅ ⋅ ⋅ ⋅ ( 2 m + 1 ) Calculation : Step 1: Using the formulas and substituting the values of m as 8,9,10, we get A 8 = A 2 ⋅ 4 = π 4 4 ! = π 4 24 ≈ 4.06 A 9 = A 2 ⋅ 4 + 1 = 2 4 + 1 π 4 1 × 3 × 5 × 7 × 9 = 32 π 4 945 ≈ 3.3 A 10 = A 2 ⋅ 5 = π 5 5 ! = π 5 120 ≈ 2.55 (b) To prove : C n < 1 for n ≥ 6 Given : A unit ball in R n , C 0 = π Formula Used : C n = n − 1 n C n − 2 . Proof: Using the formula and the value C 0 = π . This gives C 6 = 6 − 1 6 C 4 = 5 6 × 4 − 1 4 C 2 = 5 8 × 2 − 1 2 C 0 = 5 π 16 ≈ 0.98 < 1 C 7 = 7 − 1 7 C 5 = 6 7 × 4 5 C 3 = 24 35 × 2 3 C 1 C 1 = ∫ − π 2 π 2 cos θ d θ = s i n θ | − π 2 π 2 = 2 C 7 = 16 35 × 2 = 32 35 < 1 Thus, C n < 1 for n = 6 a n d 7 . We now assume that C k < 1 for all k, 6 ≤ k ≤ n for some n ≥ 7 . We show that C n + 1 < 1 . (c) To prove : Of all the unit balls, the five dimensional ball has the largest volume and explain why when n → ∞ , A n → 0. Given : A unit ball in R n , C 0 = π , C n < 1 for n ≥ 6 Formula Used : C n = n − 1 n C n − 2 Proof: Using the formula , we get C n + 1 = ( n + 1 ) − 1 n + 1 C ( n + 1 ) − 2 = n n + 1 C n − 1 By the assumption, n − 1 ≥ 6 , and so C n − 1 < 1 ...

4th Edition

ISBN: 9781319434526

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.3, Problem 46E

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(a) To calculate: The volume $A_{n}$ of the unit ball in $R^{n}$ for n=8,9,10

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Chapter 16.3, Problem 45E

Chapter 16.4, Problem 1PQ

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Chapter 16 Solutions

CALCULUS 4E (LL) W/ ACHIEVE ACCESS

Ch. 16.1 - Prob. 1PQ Ch. 16.1 - Prob. 2PQ Ch. 16.1 - Prob. 3PQ Ch. 16.1 - Prob. 4PQ Ch. 16.1 - Prob. 5PQ Ch. 16.1 - Prob. 6PQ Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E

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(a) To calculate : The volume A n of the unit ball in R n for n=8,9,10

Solution Summary: The author explains how to calculate the volume of the unit ball in R n, using the formulas and the value C 0 =.

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(a) To calculate: The volume $A_{n}$ of the unit ball in $R^{n}$ for n=8,9,10

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Chapter 16 Solutions

(a) To calculate : The volume A n of the unit ball in R n for n=8,9,10

Solution Summary: The author explains how to calculate the volume of the unit ball in R n, using the formulas and the value C 0 =.

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(a) To calculate: The volume An of the unit ball in Rn for n=8,9,10

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Chapter 16 Solutions

(a) To calculate: The volume $A_{n}$ of the unit ball in $R^{n}$ for n=8,9,10