Explanation Given information: A needle of length 1 unit is dropped on a floor that is ruled with parallel lines 1 unit apart. Consider P to be the lowest point of the needle, y the distance of P from the ruled line above it, and θ the angle the needle makes with a line parallel to the ruled lines. Formula used: Area of rectangle = Length × Width . Proof: The length of the needle is 1 unit, and the angle made by the needle with the ruled line is θ . The component of P in polar coordinate is ( r cos θ , r sin θ ) , where r is the length of the needle. Therefore, the vertical component of the needle is sin θ . Variable y is the distance between the base of the needle and the ruled line above it. The needle will touch the above ruled line if and only if the distance y will be less than or equal to the vertical component of the needle, that is, sin θ . That means, the needle will touch the above ruled line if and only if y ≤ sin θ . Distance y follows uniform distribution with 0 ≤ y ≤ 1 and 0 ≤ θ ≤ π with probability density function 1 π . If a needle is dropped, it will be at any point ( θ , y ) satisfying the function y = sin θ , 0 ≤ θ ≤ π . It will be the point on the rectangle with 0 ≤ y ≤ 1 and 0 ≤ θ ≤ π . Thus, Pr ( y ≤ sin θ ) = ∫ 0 π ∫ 0 sin θ 1 π d y d θ

Student Solutions Manual for Calculus & Its Applications and Calculus & Its Applications, Brief Version

14th Edition

ISBN: 9780134463230

Author: Larry J. Goldstein, David I Lay, David I. Schneider, Nakhle H. Asmar

Publisher: PEARSON

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Chapter 12, Problem 16RE

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To prove: The Buffon needle touches a ruled line, if and only if, y ≤ sin θ.

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Chapter 12, Problem 15RE

Chapter 12, Problem 17RE

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Chapter 12 Solutions

Student Solutions Manual for Calculus & Its Applications and Calculus & Its Applications, Brief Version

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Ch. 12.1 - Decision Making Based on Expected Value A citrus...Ch. 12.1 - Prob. 10E Ch. 12.2 - Prob. 1CYU Ch. 12.2 - Prob. 2CYU Ch. 12.2 - Prob. 1E Ch. 12.2 - Prob. 2E Ch. 12.2 - Prob. 3E Ch. 12.2 - Prob. 4E Ch. 12.2 - Prob. 5E Ch. 12.2 - Prob. 6E Ch. 12.2 - Prob. 7E Ch. 12.2 - Prob. 8E Ch. 12.2 - Prob. 9E Ch. 12.2 - Prob. 10E Ch. 12.2 - Prob. 11E Ch. 12.2 - Prob. 12E Ch. 12.2 - Prob. 13E Ch. 12.2 - Prob. 14E Ch. 12.2 - Prob. 15E Ch. 12.2 - Prob. 16E Ch. 12.2 - Prob. 17E Ch. 12.2 - Prob. 18E Ch. 12.2 - Prob. 19E Ch. 12.2 - Prob. 20E Ch. 12.2 - Prob. 21E Ch. 12.2 - Prob. 22E Ch. 12.2 - Prob. 23E Ch. 12.2 - Prob. 24E Ch. 12.2 - Prob. 25E Ch. 12.2 - Prob. 26E Ch. 12.2 - An experiment consists of selecting a point at...Ch. 12.2 - Prob. 28E Ch. 12.2 - Prob. 29E Ch. 12.2 - Prob. 30E Ch. 12.2 - Prob. 31E Ch. 12.2 - Prob. 32E Ch. 12.2 - Prob. 33E Ch. 12.2 - Prob. 34E Ch. 12.2 - Prob. 35E Ch. 12.2 - A random variable X has a cumulative distribution...Ch. 12.2 - Prob. 37E Ch. 12.2 - Prob. 38E Ch. 12.3 - 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To prove: The Buffon needle touches a ruled line, if and only if, y ≤ sin θ .

Solution Summary: The author explains how the Buffon needle touches a ruled line if the distance between the base of the needle and the above-ruled line is less than or equal to the vertical component.

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To prove: The Buffon needle touches a ruled line, if and only if, y ≤ sin θ.

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Chapter 12 Solutions