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Calculate the distance between the center of gravity of the Earth-Moon system and the center of the Earth. 

Mass of the Earth (mE) = 5.98 × 1024kg

Mass of the Moon (mM) = 7.35 × 1022kg

Radius of the orbit rM = 384.4 × 103 km of the moon

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Let x be the distance between the center of the Earth and the center of gravity of the Earth-Moon system. Taking moments around the center of gravity: 

5.98 × 1024 (x) = 7.35 × 1022 (384.4 × 103 - x)

Solving for x:

5,98 × 1024 (x) = 2825.3 × 1025 - 7.35 × 1022 (x)

598 × 1022(x) + 7.35 × 1022(x) = 2825.3 × 1025 

605.35 × 1022(x) = 2825.3 × 1025

x = (2825.3 x 1025)/(605.35 x 1022)

x = 4.667 x 103km

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