Explanation Given: Length of minute hand of a clock = 8 cm Length of hour hand of a clock = 5 cm Formula used: Law of Cosines: By the figure, the formula is: c 2 = a 2 + b 2 − 2 a b cos C Calculation: The hour hand travels around the clock: 1 rev 12 h = 2 π radian 12 h = π radian 6 h The minute hand travels around the clock: 1 rev 1 h = 2 π radian 1 h = 2 π radian h Let the θ between them change at d θ d t = π 6 − 2 π = − 11 π radian 6 h Let D be the distance between the tips of the hands. So, d D d t at 3 o’clock. θ = 3 12 ⋅ 2 π = π 2 Now, use the Law of Cosines. D 2 = 5 2 + 8 2 − 2 ( 5 ) ( 8 ) cos θ D 2 = 89 − 80 cos θ … (1) Find the derivative of above function with respect to t . d d t ( D 2 ) = d d t ( 89 − 80 cos θ ) 2 D d D d t = 80 sin θ d θ d t Divide both sides by 2 D . 2 D 2 D d D d t = 80 sin θ 2 D d θ d t d D d t = 40 sin θ D d θ d t … (2) Substitute θ = π 2 in the equation (1)

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 3, Problem 81CRE

To determine

How fast is the distance between the tips of the hands changing at 3 o’clock?

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Chapter 3, Problem 80CRE

Chapter 3, Problem 82CRE

Chapter 3 Solutions

Calculus - Standalone book

Ch. 3.1 - Prob. 1PQ Ch. 3.1 - Prob. 2PQ Ch. 3.1 - Prob. 3PQ Ch. 3.1 - Prob. 4PQ Ch. 3.1 - Prob. 5PQ Ch. 3.1 - Prob. 1E Ch. 3.1 - Prob. 2E Ch. 3.1 - Prob. 3E Ch. 3.1 - Prob. 4E Ch. 3.1 - Prob. 5E

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How fast is the distance between the tips of the hands changing at 3 o’clock?

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