Explanation Given: The latitude and longitude of Los Angeles is respectively 34.06 ° N and 118.25 ° W . The latitude and longitude of Montreal is respectively 45.50 ° N and 73.60 ° W . The latitude of the point P is α = 90 ° − ϕ ° and the longitude of the point P is β = 360 ° − θ ° . The radius of the earth is 3960 mi. Formula used: The great circle distance between any two coordinate of three dimensional space is given by, d = σ × π × ρ 180 ° . Here, the angle between the given points is given by σ , the radius of the Earth is given by ρ (1) The dot product of two given points is A . B = | A | × | B | × cos ( σ ) (2) The spherical coordinates ( ρ , θ , ϕ ) corresponding to the rectangular coordinates ( x , y , z ) is, ρ = x 2 + y 2 + z 2 ϕ = cos − 1 ( z ρ ) θ = cos − 1 ( x ρ sin ϕ ) Calculation: Let A and B represents the three dimensional coordinates of the Los Angeles and Montreal respectively that is A = ( x 1 , y 1 , z 1 ) , B = ( x 2 , y 2 , z 2 ) . Let the radius of the earth be denoted by ρ . Convert the coordinates into spherical coordinates by the above mentioned formula. Thus, A = ( ρ sin ϕ 1 cos θ 1 , ρ sin ϕ 1 sin θ 1 , ρ cos ϕ 1 ) and B = ( ρ sin ϕ 2 cos θ 2 , ρ sin ϕ 2 sin θ 2 , ρ cos ϕ 2 ) . ρ is not varying because it denotes the radius of the earth. With the given conditions, find the corresponding latitude and longitude for the point P as shown below. α = 90 ° − ϕ 1 ° 34.06 ° = 90 ° − ϕ 1 ° 34.06 ° − 90 ° = − ϕ 1 ° ϕ 1 ° = 55.94 ° α = 90 ° − ϕ 2 ° 45.50 ° = 90 ° − ϕ 2 ° 45.50 ° − 90 ° = − ϕ 2 ° ϕ 2 ° = 44

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ISBN: 9781133710882

Author: Stewart

Publisher: CENGAGE LEARNING - CONSIGNMENT

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Cylindrical Coordinate System

The distance from a selected reference axis, the direction from the axis relative to a selected reference direction, and also the distance from a selected reference plane perpendicular to the axis area unit, all accustom outline purpose positions in a cy…

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Chapter 12.7, Problem 42E

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To find: The great circle distance from Los Angeles to Montreal.

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Chapter 12.7, Problem 41E

Chapter 12.7, Problem 44E

Chapter 12 Solutions

EBK ESSENTIAL CALCULUS: EARLY TRANSCEND

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The great circle distance from Los Angeles to Montreal.

Solution Summary: The author calculates the great circle distance from Los Angeles to Montreal, which is approximately 2464 mi.

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To find: The great circle distance from Los Angeles to Montreal.

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Chapter 12 Solutions