ps11sol

# ps11sol - 18.02 Fall 2008 – Problem Set 11 Part B...

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Unformatted text preview: 18.02 Fall 2008 – Problem Set 11, Part B Solutions 1. a) North of 42 ◦ N is 0 ≤ φ < φ , where φ = (90 − 42) π/ 180 radians. integraltext 2 π integraltext φ sin φ dφdθ integraltext 2 π integraltext π sin φ dφdθ = integraltext φ sin φ dφ integraltext π sin φ dφ = − cos φ − ( − 1) 2 ≈ . 165 About 16.5 percent of the Earth’s surface is north of Boston. b) First take the average of φ : ¯ φ = integraltext 2 π integraltext π π/ 2 φ sin φ dφdθ integraltext 2 π integraltext π π/ 2 sin φ dφdθ = π − 1 because integraltext π π/ 2 sin φ dφ = 1, and, by integration by parts, integraldisplay π π/ 2 φ sin φ dφ = − integraldisplay π π/ 2 ( − cos φ ) dφ + ( − cos φ ) φ vextendsingle vextendsingle vextendsingle π π/ 2 = π + integraldisplay π π/ 2 cos φ dφ = π − 1 Thus ¯ φ = π − 1 radian and the average latitude is 32 . 7 ◦ S. The closest large cities are Santiago, Chile (33 . 4 ◦ S) and Perth, Australia (31 .....
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ps11sol - 18.02 Fall 2008 – Problem Set 11 Part B...

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