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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 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 ?

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Calculus: Early Transcendentals

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285741550
BuyFind

Calculus: Early Transcendentals

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285741550

Solutions

Chapter
Section
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, Lh, 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 ≤ yL and 0 ≤ θπ. 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 ≤ yL, 0 ≤ θπ, 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 ?

Chapter 8, Problem 11P, In a famous 18th-century problem. known as Buffons needle problem, a needle of length h is dropped

Expert Solution
To determine

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

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

Explanation of Solution

Given information:

The needle intersects a line in the ratio of P=areaundery=hsinθareaofrectangle (1)

The limits are 0yL,0θπ

Calculation:

Find the probability of the needle to intersect a line for h=L :

Substitute L for h in Equation (1).

P=areaundery=Lsinθareaofrectangle (2)

The area under y=Lsinθ , for limits of 0θπ is expressed as,

Areaundery=Lsinθ=0πLsinθdθ

The area of rectangle is πL .

Substitute 0πLsinθdθ for areaundery=Lsinθ and πL for areaofrectangle in Equation (2).

P=0πLsinθdθπL=0πsinθdθπ=[cosθ]0ππ=cos(π)+cos(0)π

=1+1π=2π

Therefore, the probability of the needle to intersect a line h=L is 2π_

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Chapter 8 Solutions

Calculus: Early Transcendentals
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Ch. 8.1 - Find the exact length of the curve. 11....Ch. 8.1 - Find the exact length of the curve. 12....Ch. 8.1 - Find the exact length of the curve. 13....Ch. 8.1 - Find the exact length of the curve. 14. y = ln(cos...Ch. 8.1 - Find the exact length of the curve. 15. y = ln(sec...Ch. 8.1 - Find the exact length of the curve. 16....Ch. 8.1 - Find the exact length of the curve. 17....Ch. 8.1 - Find the exact length of the curve. 18....Ch. 8.1 - Find the exact length of the curve. 19....Ch. 8.1 - Find the exact length of the curve. 20. y = 1 ex,...Ch. 8.1 - Find the length of the arc of the curve from point...Ch. 8.1 - Find the length of the arc of the curve from point...Ch. 8.1 - Graph the curve and visually estimate its length....Ch. 8.1 - Graph the curve and visually estimate its length....Ch. 8.1 - Use Simpsons Rule with n = 10 to estimate the arc...Ch. 8.1 - Use Simpsons Rule with n = 10 to estimate the arc...Ch. 8.1 - Use Simpsons Rule with n = 10 to estimate the arc...Ch. 8.1 - Use Simpsons Rule with n = 10 to estimate the arc...Ch. 8.1 - (a) Graph the curve y=x4x3, 0x4. (b) Compute the...Ch. 8.1 - Repeat Exercise 29 for the curve y=x+sinx0x2Ch. 8.1 - Sketch the curve with equation x2/3 + y2/3 = 1 and...Ch. 8.1 - (a) Sketch the curve y3 = x2. (b) Use Formulas 3...Ch. 8.1 - Find the arc length function for the curve y =...Ch. 8.1 - (a) Find the arc length function for the curve y =...Ch. 8.1 - Find the arc length function for the curve...Ch. 8.1 - The arc length function for a curve y = f(x),...Ch. 8.1 - For the function f(x)=14ex+ex, prove that the arc...Ch. 8.1 - A steady wind blows a kite due west. The kites...Ch. 8.1 - A hawk flying at 15m/s at an altitude of 180 m...Ch. 8.1 - The Gateway Arch in St. Louis (see the photo on...Ch. 8.1 - A manufacturer of corrugated metal roofing wants...Ch. 8.1 - (a) The figure shows a telephone wire hanging...Ch. 8.1 - Find the length of the curve y=1xt31dt1x4Ch. 8.1 - The curves with equations xn + yn = 1, n = 4, 6,...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - The given curve is rotated about the y-axis. Find...Ch. 8.2 - The given curve is rotated about the y-axis. Find...Ch. 8.2 - The given curve is rotated about the y-axis. Find...Ch. 8.2 - The given curve is rotated about the y-axis. Find...Ch. 8.2 - Use Simpsons Rule with n = 10 to approximate the...Ch. 8.2 - Use Simpsons Rule with n = 10 to approximate the...Ch. 8.2 - Use Simpsons Rule with n = 10 to approximate the...Ch. 8.2 - Use Simpsons Rule with n = 10 to approximate the...Ch. 8.2 - Use either a CAS or a table of integrals to find...Ch. 8.2 - Use either a CAS or a table of integrals to find...Ch. 8.2 - If the region = {(x, y) | x 1, 0 y 1/x} is...Ch. 8.2 - If the infinite curve y = ex, x 0, is rotated...Ch. 8.2 - (a) If a 0, find the area of the surface...Ch. 8.2 - A group of engineers is building a parabolic...Ch. 8.2 - (a) The ellipse x2a2+y2b2=1ab is rotated about the...Ch. 8.2 - Find the surface area of the torus in Exercise...Ch. 8.2 - If the curve y = f(x), a x b, is rotated about...Ch. 8.2 - Find the area of the surface obtained by rotating...Ch. 8.2 - (a) Show that the surface area of a zone of a...Ch. 8.2 - Show that if we rotate the curve y = ex/2 + e x/2...Ch. 8.2 - Let L be the length of the curve y = f(x), a x ...Ch. 8.2 - Formula 4 is valid only when f(x) 0. Show that...Ch. 8.3 - An aquarium 5 ft long, 2 ft wide, and 3 ft deep is...Ch. 8.3 - A tank is 8 m long, 4 m wide, 2 m high, and...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A milk truck carries milk with density 64.6 lb/ft3...Ch. 8.3 - A trough is filled with a liquid of density 840...Ch. 8.3 - A vertical dam has a semicircular gate as shown in...Ch. 8.3 - A cube with 20-cm-long sides is sitting on the...Ch. 8.3 - A dam is inclined at an angle of 30 from the...Ch. 8.3 - A swimming pool is 20 ft wide and 40 ft long and...Ch. 8.3 - Suppose that a plate is immersed vertically in a...Ch. 8.3 - A metal plate was found submerged vertically in...Ch. 8.3 - (a) Use the formula of Exercise 18 to show that...Ch. 8.3 - Point-masses mi arc located on the x-axis as...Ch. 8.3 - Point-masses mi arc located on the x-axis as...Ch. 8.3 - The masses mi, are located at the points Pi. Find...Ch. 8.3 - The masses mi, are located at the points Pi. Find...Ch. 8.3 - Sketch the region bounded by the curves, and...Ch. 8.3 - Sketch the region bounded by the curves, and...Ch. 8.3 - Sketch the region bounded by the curves, and...Ch. 8.3 - Sketch the region bounded by the curves, and...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Calculate the moments Mx and My and the center of...Ch. 8.3 - Calculate the moments Mx and My and the center of...Ch. 8.3 - Use Simpsons Rule to estimate the centroid of the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Use a graph to find approximate x-coordinates of...Ch. 8.3 - Prove that the centroid of any triangle is located...Ch. 8.3 - Find the centroid of the region shown, not by...Ch. 8.3 - Find the centroid of the region shown, not by...Ch. 8.3 - A rectangle with sides a and b is divided into...Ch. 8.3 - If x is the x-coordinate of the centroid of the...Ch. 8.3 - Use the Theorem of Pappus to find the volume of...Ch. 8.3 - Use the Theorem of Pappus to find the volume of...Ch. 8.3 - Use the Theorem of Pappus to find the volume of...Ch. 8.3 - The centroid of a curve can be found by a process...Ch. 8.3 - The Second Theorem of Pappus is in the same spirit...Ch. 8.3 - Use the Second Theorem of Pappus described in...Ch. 8.3 - Let be the region that lies between the curves...Ch. 8.3 - Prove Formulas 9.Ch. 8.4 - The marginal cost function C(x) was defined to be...Ch. 8.4 - A company estimates that the marginal revenue (in...Ch. 8.4 - A mining company estimates that the marginal cost...Ch. 8.4 - The demand function for a particular vacation...Ch. 8.4 - A demand curve is given by p = 450/(x + 8). Find...Ch. 8.4 - The supply function ps(x) for a commodity gives...Ch. 8.4 - If a supply curve is modeled by the equation p =...Ch. 8.4 - In a purely competitive market, the price of a...Ch. 8.4 - The sum of consumer surplus and producer surplus...Ch. 8.4 - A camera company estimates that the demand...Ch. 8.4 - A company modeled the demand curve for its product...Ch. 8.4 - A movie theater has been charging 10.00 per...Ch. 8.4 - If the amount of capital that a company has at...Ch. 8.4 - If revenue flows into a company at a rate of...Ch. 8.4 - If income is continuously collected at a rate of...Ch. 8.4 - The present value of an income stream is the...Ch. 8.4 - Paretos Law of Income states that the number of...Ch. 8.4 - A hot, wet summer is causing a mosquito population...Ch. 8.4 - Use Poiseuilles Law to calculate the rate of flow...Ch. 8.4 - High blood pressure results from constriction of...Ch. 8.4 - The dye dilution method is used to measure cardiac...Ch. 8.4 - After a 5.5-mg injection of dye, the readings of...Ch. 8.4 - The graph of the concentration function c(t) is...Ch. 8.5 - Let f(x) be the probability density function for...Ch. 8.5 - Let f(t) be the probability density function for...Ch. 8.5 - Let .f(x) = 30x2(1 x)2 for 0 x 1 and f(x) = 0...Ch. 8.5 - The density function f(x)=e3x(1+e3x)2 is an...Ch. 8.5 - Let f(x) = c/(1 + x2). (a) For what value of c is...Ch. 8.5 - Let f(x) = k(3x x2) if 0 x 3 and f(x) = 0 if x ...Ch. 8.5 - A spinner from a board game randomly indicates a...Ch. 8.5 - (a) Explain why the function whose graph is shown...Ch. 8.5 - Show that the median waiting time for a phone call...Ch. 8.5 - (a) A type of light bulb is labeled as having an...Ch. 8.5 - An online retailer has determined that the average...Ch. 8.5 - The time between infection and the display of...Ch. 8.5 - REM sleep is the phase of sleep when most active...Ch. 8.5 - According to the National Health Survey, the...Ch. 8.5 - The Garbage Project at the University of Arizona...Ch. 8.5 - Boxes are labeled as containing 500 g of cereal....Ch. 8.5 - The speeds of vehicles on a highway with speed...Ch. 8.5 - Show that the probability density function for a...Ch. 8.5 - For any normal distribution, find the probability...Ch. 8.5 - The standard deviation for a random variable with...Ch. 8.5 - The hydrogen atom is composed of one proton in the...Ch. 8 - (a) How is the length of a curve defined? (b)...Ch. 8 - (a) Write an expression for the surface area of...Ch. 8 - Describe how we can find the hydrostatic force...Ch. 8 - (a) What is the physical significance of the...Ch. 8 - What does the Theorem of Pappus say?Ch. 8 - Given a demand function p(x), explain what is...Ch. 8 - (a) What is the cardiac output of the heart? (b)...Ch. 8 - What is a probability density function? What...Ch. 8 - Suppose f(x) is the probability density function...Ch. 8 - What is a normal distribution? What is the...Ch. 8 - Find the length of the curve. y = 4(x 1)3/2, 1 x...Ch. 8 - Find the length of the curve. y=2ln(sin12x),/3xCh. 8 - Find the length of the curve. 12x = 4y3 + 3y1, 1 ...Ch. 8 - (a) Find the length of the curve y=x416+12x21x2...Ch. 8 - Let C be the arc of the curve y = 2/ (x + 1) from...Ch. 8 - (a) The curve y = x2, 0 x 1, is rotated about...Ch. 8 - Use Simpson's Rule with n = 10 to estimate the...Ch. 8 - Use Simpsons Rule with n = 10 to estimate the area...Ch. 8 - Find the length of the curve y=1xt1dt1x16Ch. 8 - What is a normal distribution? What is the...Ch. 8 - A gate in an irrigation canal is constructed in...Ch. 8 - A trough is filled with water and its vertical...Ch. 8 - Find the centroid of the region shown.Ch. 8 - Find the centroid of the region shown.Ch. 8 - Find the centroid of the region bounded by the...Ch. 8 - Find the centroid of the region bounded by the...Ch. 8 - Find the volume obtained when the circle of radius...Ch. 8 - Use the Theorem of Pappus and the fact that the...Ch. 8 - The demand function for a commodity is given by p...Ch. 8 - After a 6-mg injection of dye into a heart, the...Ch. 8 - (a) Explain why the function...Ch. 8 - Lengths of human pregnancies are normally...Ch. 8 - The length of time spent waiting in line at a...Ch. 8 - Find the area of the region S = {(x, y) | x 0, y ...Ch. 8 - Find the centroid of the region enclosed by the...Ch. 8 - If a sphere of radius r is sliced by a plane whose...Ch. 8 - (a) Show that an observer at height H above the...Ch. 8 - Suppose that the density of seawater, = (z),...Ch. 8 - The figure shows a semicircle with radius 1,...Ch. 8 - Let P be a pyramid with a square base of side 2b...Ch. 8 - Consider a flat metal plate to be placed...Ch. 8 - A uniform disk with radius 1 m is to be cut by a...Ch. 8 - A triangle with area 30 cm2 is cut from a corner...Ch. 8 - In a famous 18th-century problem. known as Buffons...Ch. 8 - If the needle in Problem 11 has length h L, its...Ch. 8 - Find the centroid of the region enclosed by the...

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