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Two crystal spheres of diameter x/2 are being packed in a cubic box wi

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Bunuel
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Two crystal spheres of diameter x/2 are being packed in a cubic box wi [#permalink] New post   16 May 2022, 01:30
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63% (02:55) correct 37% (03:05) wrong based on 43 sessions
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Two crystal spheres of diameter x/2 are being packed in a cubic box with a side of x. If the crystal spheres are in the box and the rest of the box is completely filled with packing powder, approximately what proportion of the box is filled with packing powder? (The volume of a sphere of radius r is \(\frac{4}{3}*\pi r^3\).)

(A) 17/16
(B) 1/8
(C) 1/2
(D) 3/4
(E) 7/8



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S1905
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Re: Two crystal spheres of diameter x/2 are being packed in a cubic box wi [#permalink] New post   05 Jun 2022, 07:11
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Proportion of box filled with packing powder = [Vol of Box - 2*Vol of Sphere]/Vol of Box
Vol of Box = \(x^{3}\)
Vol of Sphere = \(\frac{4}{3} * \frac{22}{7} * (\frac{x}{4})^{3} \)
Proportion of box filled with packing powder = \(73/84 \approx{7/8}\)

Ans. E
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Re: Two crystal spheres of diameter x/2 are being packed in a cubic box wi [#permalink] New post   06 Jun 2022, 09:48
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since we have sides as x and diameters as x/2 we can plug values. Taking the side to be 20 units the diameter becomes 10 units and the radius of each sphere becomes 5 units.
voulme of the cube is 20X20X 20=8000
the volume of each sphere is 4/3X3.14X5X5X5=500 (approx) for two it becomes 1000
remaining volume filled is 8000-1000=7000
or 7/8
ans E
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Re: Two crystal spheres of diameter x/2 are being packed in a cubic box wi [#permalink] New post   07 Jun 2022, 23:30
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2 spheres of diameter x/2 => radius x/4 are inside the cubic box of side x.
Volume of powder = (Volume of cube - Volume of 2 spheres)/Volume of cube = 1 - (Volume of 2 spheres/Volume of cube)
= 1 - ((2 * 4/3 * pi * (x/4)^3)/x^3)
= 1 - ((8/3 * pi)/64)
~ 1 - (pi/3*8)
= 1 - 1/8
= 7/8
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Re: Two crystal spheres of diameter x/2 are being packed in a cubic box wi [#permalink]
07 Jun 2022, 23:30
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