Explanation The given acceleration of the piston is a = 2 w 2 ( cos θ + 0.5 cos 2 θ ) . Obtain the derivative of a as follows. d a d θ = d d θ ( 2 w 2 ( cos θ + 0.5 cos 2 θ ) ) = 2 w 2 d d θ ( cos θ + 0.5 cos 2 θ ) = 2 w 2 ( − sin θ + 0.5 ( − sin 2 θ ) ( 2 ) ) = − 2 w 2 ( sin θ + sin 2 θ ) Simplify further, d a d θ = − 2 w 2 ( sin θ + 2 sin θ cos θ ) = − 2 w 2 sin θ ( 1 + 2 cos θ ) Obtain the critical point as follows. d a d θ = 0 − 2 w 2 sin θ ( 1 + 2 cos θ ) = 0 sin θ ( 1 + 2 cos θ ) = 0 sin θ = 0 or 1 + 2 cos θ = 0 Simplify further, sin θ = 0 or cos θ = − 1 2 θ = 0 , π or θ = 2 π 3 , 4 π 3 Obtain the second derivative of a as follows. d 2 a d θ 2 = d d θ ( d a d θ ) = d d θ ( − 2 w 2 sin θ ( 1 + 2 cos θ ) ) = − 2 w 2 d d θ ( sin θ ( 1 + 2 cos θ ) ) = − 2 w 2 [ sin θ d d θ ( 1 + 2 cos θ ) + ( 1 + 2 cos θ ) d d θ ( sin θ ) ] Simplify further, d 2 a d θ 2 = − 2 w 2 [ sin θ ( 0 − 2 sin θ ) + ( 1 + 2 cos θ ) ( cos θ ) ] = − 2 w 2 [ − 2 sin 2 θ + cos θ + 2 cos 2 θ ] = − 2 w 2 [ 2 ( cos 2 θ − sin 2 θ ) + cos θ ] = − 2 w 2 [ 2 cos 2 θ + cos θ ] At θ = 0 , the acceleration is: Substitute θ = 0 in d 2 a d θ 2 = − 2 w 2 [ 2 cos 2 θ + cos θ ] . d 2 a d θ 2 = − 2 w 2 [ 2 cos 2 ( 0 ) + cos ( 0 ) ] = − 2 w 2 [ 2 ( 1 ) + ( 1 ) ] = − 6 w 2 < 0 Thus, the acceleration is locally maximized. At θ = 2 π 3 , the acceleration is: Substitute θ = 2 π 3 in d 2 a d θ 2 = − 2 w 2 [ 2 cos 2 θ + cos θ ] . d 2 a d θ 2 = − 2 w 2 [ 2 cos 2 ( 2 π 3 ) + cos ( 2 π 3 ) ] = − 2 w 2 [ 2 ( − 1 2 ) + ( − 1 2 ) ] = − 2 w 2 [ − 3 2 ] = 3 w 2 > 0 Thus, the acceleration is locally minimized

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ISBN: 9780134437705

Author: Washington

Publisher: PEARSON

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Chapter 27.8, Problem 45E

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The value of θ for maximum or minimum a.

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Chapter 27.8, Problem 43E

Chapter 27.8, Problem 46E

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Q1. A chest of drawers has 3 drawers. Each drawer has 2 boxes. The boxes of one drawer contain a silver coin in each respectively, the boxes of another a gold coin in each box, and the boxes of the third drawer a gold and a silver coin, respectively. A drawer is selected at random and a box from the drawer is selected at random and opened. The coin is found to be silver. What is the probability that the coin in the other box is gold? (Harder Problem)

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Chapter 27 Solutions

Basic Technical Mathematics

Ch. 27.1 - Prob. 1PE Ch. 27.1 - Prob. 2PE Ch. 27.1 - Prob. 1E Ch. 27.1 - Prob. 2E Ch. 27.1 - Prob. 3E Ch. 27.1 - Prob. 4E Ch. 27.1 - Prob. 5E Ch. 27.1 - Prob. 6E Ch. 27.1 - Prob. 7E Ch. 27.1 - Prob. 8E

Ch. 27.1 - Prob. 9E Ch. 27.1 - Prob. 10E Ch. 27.1 - Prob. 11E Ch. 27.1 - Prob. 12E Ch. 27.1 - Prob. 13E Ch. 27.1 - Prob. 14E Ch. 27.1 - Prob. 15E Ch. 27.1 - Prob. 16E Ch. 27.1 - Prob. 17E Ch. 27.1 - Prob. 18E Ch. 27.1 - Prob. 19E Ch. 27.1 - Prob. 20E Ch. 27.1 - Prob. 21E Ch. 27.1 - Prob. 22E Ch. 27.1 - Prob. 23E Ch. 27.1 - Prob. 24E Ch. 27.1 - Prob. 25E Ch. 27.1 - In Exercises 3–34, find the derivatives of the...Ch. 27.1 - Prob. 27E Ch. 27.1 - Prob. 28E Ch. 27.1 - Prob. 29E Ch. 27.1 - Prob. 30E Ch. 27.1 - Prob. 31E Ch. 27.1 - Prob. 32E Ch. 27.1 - Prob. 33E Ch. 27.1 - Prob. 34E Ch. 27.1 - Prob. 35E Ch. 27.1 - Prob. 36E Ch. 27.1 - Prob. 37E Ch. 27.1 - Prob. 38E Ch. 27.1 - Prob. 39E Ch. 27.1 - Prob. 40E Ch. 27.1 - Prob. 41E Ch. 27.1 - Prob. 42E Ch. 27.1 - Prob. 43E Ch. 27.1 - Prob. 44E Ch. 27.1 - Prob. 45E Ch. 27.1 - Prob. 46E Ch. 27.1 - Prob. 47E Ch. 27.1 - Prob. 48E Ch. 27.1 - Prob. 49E Ch. 27.1 - Prob. 50E Ch. 27.1 - Prob. 51E Ch. 27.1 - Prob. 52E Ch. 27.1 - Prob. 53E Ch. 27.1 - Prob. 54E Ch. 27.1 - The blade of a saber saw moves vertically up and...Ch. 27.1 - Prob. 56E Ch. 27.1 - Prob. 57E Ch. 27.1 - Prob. 58E Ch. 27.1 - Prob. 59E Ch. 27.1 - Prob. 60E Ch. 27.2 - Prob. 1PE Ch. 27.2 - Prob. 2PE Ch. 27.2 - Prob. 1E Ch. 27.2 - Prob. 2E Ch. 27.2 - Prob. 3E Ch. 27.2 - Prob. 4E Ch. 27.2 - Prob. 5E Ch. 27.2 - Prob. 6E Ch. 27.2 - Prob. 7E Ch. 27.2 - Prob. 8E Ch. 27.2 - Prob. 9E Ch. 27.2 - Prob. 10E Ch. 27.2 - Prob. 11E Ch. 27.2 - Prob. 12E Ch. 27.2 - Prob. 13E Ch. 27.2 - Prob. 14E Ch. 27.2 - Prob. 15E Ch. 27.2 - Prob. 16E Ch. 27.2 - Prob. 17E Ch. 27.2 - Prob. 18E Ch. 27.2 - Prob. 19E Ch. 27.2 - Prob. 20E Ch. 27.2 - Prob. 21E Ch. 27.2 - Prob. 22E Ch. 27.2 - Prob. 23E Ch. 27.2 - Prob. 24E Ch. 27.2 - Prob. 25E Ch. 27.2 - Prob. 26E Ch. 27.2 - Prob. 27E Ch. 27.2 - Prob. 28E Ch. 27.2 - Prob. 29E Ch. 27.2 - Prob. 30E Ch. 27.2 - Prob. 31E Ch. 27.2 - Prob. 32E Ch. 27.2 - Prob. 33E Ch. 27.2 - In Exercises 3–34, find the derivatives of the...Ch. 27.2 - Prob. 35E Ch. 27.2 - Prob. 36E Ch. 27.2 - Prob. 37E Ch. 27.2 - 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Prob. 7E Ch. 27.7 - Prob. 8E Ch. 27.7 - Prob. 9E Ch. 27.7 - Prob. 10E Ch. 27.7 - Prob. 11E Ch. 27.7 - Prob. 12E Ch. 27.7 - Prob. 13E Ch. 27.7 - Prob. 14E Ch. 27.7 - Prob. 15E Ch. 27.7 - Prob. 16E Ch. 27.7 - Prob. 17E Ch. 27.7 - Prob. 18E Ch. 27.7 - Prob. 19E Ch. 27.7 - Prob. 20E Ch. 27.7 - Prob. 21E Ch. 27.7 - Prob. 22E Ch. 27.7 - Prob. 23E Ch. 27.7 - Prob. 24E Ch. 27.7 - Prob. 25E Ch. 27.7 - Prob. 26E Ch. 27.7 - Prob. 27E Ch. 27.7 - Prob. 28E Ch. 27.7 - Prob. 29E Ch. 27.7 - In Exercises 3–36, evaluate each limit (if it...Ch. 27.7 - Prob. 31E Ch. 27.7 - Prob. 32E Ch. 27.7 - Prob. 33E Ch. 27.7 - Prob. 34E Ch. 27.7 - Prob. 35E Ch. 27.7 - Prob. 36E Ch. 27.7 - Prob. 37E Ch. 27.7 - Prob. 38E Ch. 27.7 - Prob. 39E Ch. 27.7 - Prob. 40E Ch. 27.7 - Prob. 41E Ch. 27.7 - Prob. 42E Ch. 27.7 - Prob. 43E Ch. 27.7 - Prob. 44E Ch. 27.7 - Prob. 45E Ch. 27.7 - Prob. 46E Ch. 27.7 - Prob. 47E Ch. 27.7 - Prob. 48E Ch. 27.8 - Prob. 1PE Ch. 27.8 - In Exercises 1 and 2, make the given changes in...Ch. 27.8 - Prob. 2E Ch. 27.8 - In Exercises 3–14, sketch the graphs of the given...Ch. 27.8 - Prob. 4E Ch. 27.8 - In Exercises 3–14, sketch the graphs of the given...Ch. 27.8 - Prob. 6E Ch. 27.8 - Prob. 7E Ch. 27.8 - Prob. 8E Ch. 27.8 - Prob. 9E Ch. 27.8 - Prob. 10E Ch. 27.8 - Prob. 11E Ch. 27.8 - In Exercises 3–14, sketch the graphs of the given...Ch. 27.8 - Prob. 13E Ch. 27.8 - Prob. 14E Ch. 27.8 - Prob. 15E Ch. 27.8 - Prob. 16E Ch. 27.8 - Prob. 17E Ch. 27.8 - Prob. 18E Ch. 27.8 - Prob. 19E Ch. 27.8 - Prob. 20E Ch. 27.8 - Prob. 21E Ch. 27.8 - Prob. 22E Ch. 27.8 - Prob. 23E Ch. 27.8 - Prob. 25E Ch. 27.8 - Prob. 26E Ch. 27.8 - Prob. 27E Ch. 27.8 - Prob. 28E Ch. 27.8 - Prob. 29E Ch. 27.8 - Prob. 30E Ch. 27.8 - Prob. 31E Ch. 27.8 - Prob. 32E Ch. 27.8 - Prob. 33E Ch. 27.8 - Prob. 34E Ch. 27.8 - The energy E (in J) dissipated by a certain...Ch. 27.8 - Prob. 36E Ch. 27.8 - Prob. 37E Ch. 27.8 - Prob. 39E Ch. 27.8 - A package of weather instruments is propelled into...Ch. 27.8 - Prob. 41E Ch. 27.8 - Prob. 42E Ch. 27.8 - Prob. 43E Ch. 27.8 - Prob. 45E Ch. 27.8 - Prob. 46E Ch. 27 - Prob. 1RE Ch. 27 - Prob. 2RE Ch. 27 - Prob. 3RE Ch. 27 - Prob. 4RE Ch. 27 - Prob. 5RE Ch. 27 - Prob. 6RE Ch. 27 - Prob. 7RE Ch. 27 - Prob. 8RE Ch. 27 - Prob. 9RE Ch. 27 - Prob. 10RE Ch. 27 - Prob. 11RE Ch. 27 - Prob. 12RE Ch. 27 - Prob. 13RE Ch. 27 - Prob. 14RE Ch. 27 - Prob. 15RE Ch. 27 - Prob. 16RE Ch. 27 - Prob. 17RE Ch. 27 - Prob. 18RE Ch. 27 - Prob. 19RE Ch. 27 - Prob. 20RE Ch. 27 - Prob. 21RE Ch. 27 - Prob. 22RE Ch. 27 - Prob. 23RE Ch. 27 - Prob. 24RE Ch. 27 - Prob. 25RE Ch. 27 - Prob. 26RE Ch. 27 - Prob. 27RE Ch. 27 - Prob. 28RE Ch. 27 - Prob. 29RE Ch. 27 - Prob. 30RE Ch. 27 - Prob. 31RE Ch. 27 - Prob. 32RE Ch. 27 - Prob. 33RE Ch. 27 - Prob. 34RE Ch. 27 - Prob. 35RE Ch. 27 - Prob. 36RE Ch. 27 - Prob. 37RE Ch. 27 - Prob. 38RE Ch. 27 - Prob. 39RE Ch. 27 - Prob. 40RE Ch. 27 - Prob. 41RE Ch. 27 - Prob. 42RE Ch. 27 - Prob. 43RE Ch. 27 - Prob. 44RE Ch. 27 - Prob. 45RE Ch. 27 - Prob. 46RE Ch. 27 - Prob. 47RE Ch. 27 - Prob. 48RE Ch. 27 - Prob. 49RE Ch. 27 - 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