Explanation Given: The length of the chord line is 4 cm. The length of the arc is 5 cm. Result used: (Newton’s method). (a) θ n + 1 = θ n − f ( θ n ) f ′ ( θ n ) Where f ( θ ) is function of θ . (b) s = r θ , Where s = arc length, r = radius of the circle and θ = central angle. (c) 180 ∘ = π c (d)Law of cosine cos θ = b 2 + c 2 − a 2 2 b c Calculation: From the result (b) set s = 5 and calculate radius r = 5 θ From result (d) set r = 4 , a = r and b = r . 4 2 = r 2 + r 2 − 2 × r × r × cos θ 4 2 = 2 r 2 − 2 r 2 cos θ 4 2 = 2 r 2 ( 1 − cos θ ) Set r = 5 θ to calculate the above expression. 4 2 = 2 ( 5 θ ) 2 ( 1 − cos θ ) 4 2 = 2 ( 25 θ 2 ) ( 1 − cos θ ) 16 θ 2 = 50 ( 1 − cos θ ) 16 θ 2 + 50 cos θ − 50 = 0 Let f ( θ ) = 16 θ 2 + 50 cos θ − 50 . Calculate the derivative of the above function f ( θ ) . Let the derivative of f ( θ ) is f ′ ( θ ) = d f ( θ ) d θ . f ′ ( θ ) = d ( 16 θ 2 + 50 cos θ − 50 ) d θ = 16 d ( θ 2 ) d θ + 50 d ( cos θ ) d θ − d ( 50 ) d θ = 32 θ − 50 sin θ − 0 = 32 θ − 50 sin θ Graph: From the graph the initial point θ 1 = 2.2 . The value of f ′ ( θ 1 ) at the point θ 1 = 2.2 is, f ′ ( 2.2 ) = 32 × 2.2 − 50 × sin ( 2.2 ) = 70.4 − 50 × 0.808496 = 70.4 − 40.424820 = 29.975179 Calculate the value of f ( θ 1 ) at the point θ 1 = 2

Single Variable Calculus: Early Transcendentals, Volume I

8th Edition

ISBN: 9781305270343

Author: James Stewart

Publisher: Cengage Learning

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Chapter 4.8, Problem 40E

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To Find: The central angle in radian of the circle correct to the four decimal places and give the answer to the nearest degree.

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Chapter 4.8, Problem 39E

Chapter 4.8, Problem 41E

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Chapter 4 Solutions

Single Variable Calculus: Early Transcendentals, Volume I

Ch. 4.1 - Explain the difference between an absolute minimum...Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Use the graph to state the absolute and local...Ch. 4.1 - Use the graph to state the absolute and local...Ch. 4.1 - Sketch the graph of a function f that is...Ch. 4.1 - Sketch the graph of a function f that is...Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E

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The central angle in radian of the circle correct to the four decimal places and give the answer to the nearest degree.

Solution Summary: The author explains Newton's method of finding the central angle in radian of the circle.

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To Find: The central angle in radian of the circle correct to the four decimal places and give the answer to the nearest degree.

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Chapter 4 Solutions