Explanation Given information: The volume of sphere is V = π 3 H 2 ( 3 R − H ) Where R is the radius of sphere and H is the height of the sphere. And V=8, R=2. Newton-Raphson method: x n + 1 = x n − f ( x n ) f ' ( x n ) , where n is integer. Let, H n + 1 = H n − f ( H n ) f ' ( H n ) Taking into account, H ≤ R . So, the height H ≤ 2 . Assume that the initial estimation is H 0 = 1 Differentiate the above function with respect to H. f ( H ) = π 3 H 2 ( 3 R − H ) − V f ' ( H ) = d d H ( π 3 ( 3 R H 2 − H 3 ) − V ) f ' ( H ) = ( π 3 ( 6 R H − 3 H 2 ) ) f ' ( H ) = ( π 3 ( 12 H − 3 H 2 ) ) [ Substitute R=2 ] f ' ( H ) = − π H ( H − 4 ) So, from Newton Raphson method. H n + 1 = H n − f ( H n ) f ' ( H n ) H n + 1 = H n − π 3 H n 2 ( 3 R − H n ) − V − π H n ( H n − 4 ) H n + 1 = H n − π 3 H n 2 ( 3 × 2 − H n ) − 8 − π H n ( H n − 4 ) [ ∵ V = 8 and R=2 ] H n + 1 = 24 + 6 π H n 2 − 2 π H n 3 12 π H n − 3 π H n 2 Find the value of H for n=0, 1, 2, …

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Question

Chapter 3.8, Problem 51PS

To determine

To use the Newton-Raphson method to estimate the corresponding value of H.

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Chapter 3.8, Problem 50PS

Chapter 3.8, Problem 52PS

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Chapter 3 Solutions

Calculus

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Knowledge Booster

To use the Newton-Raphson method to estimate the corresponding value of H.

Solution Summary: The author explains the Newton-Raphson method to estimate the corresponding value of H.

To use the Newton-Raphson method to estimate the corresponding value of H.

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Chapter 3 Solutions