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CalculusCalculus: Early TranscendentalsIn a famous 18th-century problem. known as Buffon’s needle problem, a needle of length h is dropped onto a flat surface (for example. a table) on which parallel lines L unit s apart, L ≥ h, have been drawn. The problem is to determine the probability that the needle will come to rest intersecting one of the lines. Assume that the lines run east-west, parallel to the x -axis in a rectangular coordinate system (as in the figure). Let y be the distance from the “southern” end of the needle to the nearest line to the north. (If the needle’s southern end lies on a line, let y = 0. If the needle happens to lie east-west, let the “western” end be the “southern” end) Let θ be the angle that the needle makes with a ray extending eastward from the “southern” end. Then 0 ≤ y ≤ L and 0 ≤ θ ≤ π . Note that the needle intersects one of the lines only when y < h sin θ . The total set of possibilities for the needle can be identified with the rectangular region 0 ≤ y ≤ L , 0 ≤ θ ≤ π , and the proportion of times that the needle intersects a line is the ratio area under y = h sin θ area of rectangle This ratio is the probability that the needle intersects a line. Find the probability that the needle will intersect a line if h = L. What if h = 1 2 L ?BuyFind*arrow_forward*

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781285741550

Chapter 8, Problem 11P

Textbook Problem

In a famous 18th-century problem. known as *Buffon’s needle problem,* a needle of length *h* is dropped onto a flat surface (for example. a table) on which parallel lines *L* unit s apart, *L* ≥ *h,* have been drawn. The problem is to determine the probability that the needle will come to rest intersecting one of the lines. Assume that the lines run east-west, parallel to the *x*-axis in a rectangular coordinate system (as in the figure). Let *y* be the distance from the “southern” end of the needle to the nearest line to the north. (If the needle’s southern end lies on a line, let *y* = 0. If the needle happens to lie east-west, let the “western” end be the “southern” end) Let *θ* be the angle that the needle makes with a ray extending eastward from the “southern” end. Then 0 ≤ *y* ≤ *L* and 0 ≤ *θ* ≤ *π*. Note that the needle intersects one of the lines only when *y* < *h* sin *θ*. The total set of possibilities for the needle can be identified with the rectangular region 0 ≤ *y* ≤ *L*, 0 ≤ *θ* ≤ *π*, and the proportion of times that the needle intersects a line is the ratio

This ratio is the probability that the needle intersects a line. Find the probability that the needle will intersect a line if *h* = *L.* What if

Expert Solution

To determine

The probability of the needle to intersect a line

The probability of the needle to intersect a line

**Given information:**

The needle intersects a line in the ratio of

The limits are

**Calculation:**

Find the probability of the needle to intersect a line for

Substitute *L* for *h* in Equation (1).

The area under

The area of rectangle is

Substitute

Therefore, the probability of the needle to intersect a line

Calculus: Early Transcendentals

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