MAT235PS2-SOLNS

# MAT235PS2-SOLNS - MAT-235 2008-2009 Problem Set 2 Solutions...

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MAT-235, 2008-2009 Problem Set 2 – Solutions Feel free to use calculators for your computations. 1. (The numbers in the following problem are only approximately correct.) Let us think of the Earth as a sphere of radius 6 , 340 km , and let us put a system of ( x,y,z ) -coordinates on it, with its axis of rotation along the z -axis and its centre at the origin. In this system of coordinates, the Royal Observatory in Greenwich, London will be along the line of the vector (1 , 0 , 1 . 255) . The ancient city of Timbuktu in Mali is 33 south of London, and at its antipodal (that is, opposite) point on the sphere is the archipelago (and republic) of Fiji . Fiji is located in the south-west Paciﬁc Ocean, and 30 east of it is the island of Tahiti , the largest and capital island of French Polynesia . If a plane were to ﬂy from London to Tahiti, what is the shortest possible distance that it would need to cover? Assume that the plane is ﬂying at a constant height of 10 km . Remarks. 1. The shortest route from a point A to a point B on a sphere is along the cross-section of the sphere by the plane passing through A , B and the centre of the sphere. 2. North-south angles always express angles between the vectors joining the points with the centre of the Earth. 3. East-west angles always express the angular displacement on a horizontal plane – that is, a plane which is perpendicular to the Earth’s axis. Solution: By remark 3, the shortest possible distance for the plane is along the arc of a circle of radius 6 , 350 km . We will compute the angle of this arc, in order to apply the formula: (Length of circular arc) = (Radius) x (Angle of arc). The vector (1 , 0 , 1 . 255) makes an angle of Arctan(1 . 255) = 51 . 45 with the positive x -axis. Since Tim- buktu is 33 south of London, it lies along the half-line of the vector (1 , 0 , tan(51 . 45 - 33 )) = (1 , 0 , 0 . 334) . Its antipodal point, Fiji, is therefore along the half-line of the vector ( - 1 , 0 , - 0 . 334) . Going to the east cor- responds to counter-clockwise rotation on the x - y -plane, when viewed from the North Pole. Therefore Tahiti, which is 30 east of Fiji, will be along the half-line of the vector ( - 1 · cos 30 , - 1 · sin 30 , - 0 . 334) = ( - 3 2 , - 1 2 , - 0 . 334) . Let a = (1 , 0 , 1 . 255) , b = ( - 3 2 , - 1 2 , - 0 . 334) . The angle θ between a and b satisﬁes: cos θ = a · b | a | · | b | . We compute:

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## This note was uploaded on 10/16/2009 for the course MATH MAT235 taught by Professor Recio during the Fall '08 term at University of Toronto.

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MAT235PS2-SOLNS - MAT-235 2008-2009 Problem Set 2 Solutions...

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