Balls

Ping pong balls have a diameter of approximately 5.1 cm. They are sold in boxes of 10 pieces: each box has the shape of a cuboid with a square base. The balls touch the walls of the box.

Calculate what portion of the internal volume of the box is filled with balls.

Result

x =  52.36 %

Solution:

x=10010V110V2=100V1V2 x=10043πr3(2r)3 x=4π10038=50π3 x=52.36%x = 100\cdot \dfrac{ 10 \cdot V_1 }{ 10 \cdot V_2 } = 100\cdot \dfrac{ V_1 }{ V_2 } \ \\ x = 100\cdot \dfrac{ \dfrac43\pi r^3 }{ (2r)^3 } \ \\ x = \dfrac{ 4\cdot \pi \cdot 100 }{ 3 \cdot 8} = \dfrac{ 50\cdot \pi }{ 3} \ \\ x = 52.36 \%



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the answer is always 52.36

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