Explanation Given information: The needle intersects a line in the ratio of P = area under y = h sin θ area of rectangle (1) The limits are 0 ≤ y ≤ L , 0 ≤ θ ≤ π Calculation: Find the probability of the needle to intersect a line for h = L : Substitute L for h in Equation (1). P = area under y = L sin θ area of rectangle (2) The area under y = L sin θ , for limits of 0 ≤ θ ≤ π is expressed as, Area under y = L sin θ = ∫ 0 π L sin θ d θ The area of rectangle is π L . Substitute ∫ 0 π L sin θ d θ for area under y = L sin θ and π L for area of rectangle in Equation (2). P = ∫ 0 π L sin θ d θ π L = ∫ 0 π sin θ d θ π = [ − cos θ ] 0 π π = − cos ( π ) + cos ( 0 ) π = 1 + 1 π = 2 π Therefore, the probability of the needle to intersect a line h = L is 2 π _

9th Edition

ISBN: 9781337613927

Author: Stewart

Publisher: CENGAGE L

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Chapter 8, Problem 11PP

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The probability of the needle to intersect a line h=L .

The probability of the needle to intersect a line h=12L .

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Chapter 8, Problem 10PP

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The probability of the needle to intersect a line h = L . The probability of the needle to intersect a line h = 1 2 L .

Solution Summary: The author calculates the probability of the needle to intersect a line for h=L.

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The probability of the needle to intersect a line h=L .

The probability of the needle to intersect a line h=12L .

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Chapter 8 Solutions

The probability of the needle to intersect a line h = L . The probability of the needle to intersect a line h = 1 2 L .

Solution Summary: The author calculates the probability of the needle to intersect a line for h=L.

Concept explainers

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The probability of the needle to intersect a line h=L . The probability of the needle to intersect a line h=12L .

Want to see the full answer?

Chapter 8 Solutions

The probability of the needle to intersect a line h=L .

The probability of the needle to intersect a line h=12L .