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705-805 Level|   Geometry|      
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A sphere is to be formed by melting smaller sphere cubes of diameter 10 Jun 2018, 02:40
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D
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A sphere is to be formed by melting smaller sphere cubes of diameter 20 mm. The sphere needs to be placed inside a cuboid of length 60 cm, width 50 cm, and height 40 cm such that it covers the maximum possible volume. How many smaller cubes are needed?

    A. 8
    B. 80
    C. 800
    D. 8000
    E. 80000


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D
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A sphere is to be formed by melting smaller sphere cubes of diameter 10 Jun 2018, 02:59
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I would believe the answer is D. Firstly, I figured that the smallest side of the cuboid is 40cm, so the final sphere must be 40cm in diameter. However, this actually does not matter. We are given answers in multiples of 8, so who cares what the actual final size really is at that scale. More importantly, we need to scale the difference. The radius of the larger sphere is 20x that of the smaller spheres. So cube that number, and we get 8000. So tada! I think.

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A sphere is to be formed by melting smaller sphere cubes of diameter 10 Jun 2018, 03:47
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Didn't get it...

Smaller spheres (or what is sphere cube?):
diameter = 20mm=2cm => R=1cm => V=Pi.

Larger sphere:
box = 40x50x60cm => for a sphere we need cube => 40x40x40cm max. Within that cube we will have a sphere of the R=20cm => V of this sphere will be 400*Pi.

So we need 400 smaller spheres, no? What am i missing here?
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A sphere is to be formed by melting smaller sphere cubes of diameter 10 Jun 2018, 03:51
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Didn't get it...

Smaller spheres (or what is sphere cube?):
diameter = 20mm=2cm => R=1cm => V=Pi.

Larger sphere:
box = 40x50x60cm => for a sphere we need cube => 40x40x40cm max. Within that cube we will have a sphere of the R=20cm => V of this sphere will be 400*Pi.

So we need 400 smaller spheres, no? What am i missing here?
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Hey worship,

volume of a sphere is calculated as \(\frac{4}{3} * pi * r^3\), where r is the radius
You may have considered that \(pi * r^2\)
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A sphere is to be formed by melting smaller sphere cubes of diameter 10 Jun 2018, 03:54
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worship
Didn't get it...

Smaller spheres (or what is sphere cube?):
diameter = 20mm=2cm => R=1cm => V=Pi.

Larger sphere:
box = 40x50x60cm => for a sphere we need cube => 40x40x40cm max. Within that cube we will have a sphere of the R=20cm => V of this sphere will be 400*Pi.

So we need 400 smaller spheres, no? What am i missing here?

Hey worship,

volume of a sphere is calculated as \(\frac{4}{3} * pi * r^3\), where r is the radius
You may have considered that \(pi * r^2\)
Show more

:lol: :lol: my bad, was thinking about the area, sorry.
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A sphere is to be formed by melting smaller sphere cubes of diameter 10 Jun 2018, 04:26
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A sphere is to be formed by melting smaller sphere cubes of diameter 20 mm. The sphere needs to be placed inside a cuboid of length 60 cm, width 50 cm, and height 40 cm such that it covers the maximum possible volume. How many smaller cubes are needed?

    A. 8
    B. 80
    C. 800
    D. 8000
    E. 80000


Articles on Geometry

Common Mistakes in Geometry Questions
Mastering Important Concepts tested by GMAT in Triangles - I
Mastering Important Concepts tested by GMAT in Triangles - II
Mastering Important Concepts tested by GMAT in Triangles - III

Consolidated list of all the articles

Must Read Articles and Practice Questions to score Q51 !!!!
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Let R and r are the radii of Big sphere and small sphere cubes respectively.

Given r=d/2=20mm/2=10mm=1cm

Given, The Big sphere needs to be placed inside a cuboid of length 60 cm, width 50 cm, and height 40 cm such that it covers the maximum possible volume. So this to happen, height of the given cuboid must be same as diameter of big sphere

Hence, Diameter of big sphere=40cm
Or, R=40/2=20cm
Given,A sphere is to be formed by melting smaller sphere cubes of diameter 20 mm.----(1)
So, r=20mm/2=10mm=1cm
Let 'n' be the no of small spherical cubes.
from (1), we have Volume of big sphere=n *Volume of one small sphere cube
Or,\(\frac{4}{3}π*R^3=n*\frac{4}{3}π*r^3\)
Or, \(\frac{4}{3}π*{20}^3=n*\frac{4}{3}π*1^3\)
Or, n=\(20^3\)=8000 nos

Ans. (D)
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A sphere is to be formed by melting smaller sphere cubes of diameter 10 Jun 2018, 06:00
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A sphere is to be formed by melting smaller sphere cubes of diameter 20 mm. The sphere needs to be placed inside a cuboid of length 60 cm, width 50 cm, and height 40 cm such that it covers the maximum possible volume. How many smaller cubes are needed?

    A. 8
    B. 80
    C. 800
    D. 8000
    E. 80000
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No idea what a sphere cube is :roll:

Let R be radius of Larger Sphere.

Given radius of smaller spheres, r = 10mm

Dimensions of the Cuboid are, L = 600mm, W = 500mm, H = 400mm

Let n be the # of smaller spheres

Volume of each smaller sphere = \(4/3{\pi}r^3\) =\(4/3{\pi}*10^3\)

Volume of larger sphere = \(4/3{\pi}R^3\)

Hence, Volume of larger sphere = \(n\) * Volume of smaller sphere

\(4/3{\pi}R^3\) = \(n * 4/3{\pi}*10^3\)

\(R^3 = n * 1000\)


Now for the maximum volume of larger sphere to fit inside the Cuboid, the diameter of the larger sphere has to be equal to the shortest dimension of the Cuboid.

Hence \(R = 400/2 = 200\) mm

So we get,

\((200)^3 = n * 1000\)

\(n = 8000\)

Answer D.


Thanks,
Gym

P.s. - Not too confident of the solution though.
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A sphere is to be formed by melting smaller sphere cubes of diameter 11 Jun 2018, 01:43
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A sphere is to be formed by melting smaller sphere cubes of diameter 20 mm. The sphere needs to be placed inside a cuboid of length 60 cm, width 50 cm, and height 40 cm such that it covers the maximum possible volume. How many smaller cubes are needed?

    A. 8
    B. 80
    C. 800
    D. 8000
    E. 80000
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Attachment:
Sphere_in_Cuboid.png
Sphere_in_Cuboid.png [ 5.44 KiB | Viewed 5800 times ]

Formula used: Volume of a sphere = \(\frac{4}{3}*\pi*R^3\)

Since the cuboid must fit the large sphere, the largest diameter possible for the sphere is 40cm(radius = 20cm)

Each small sphere needed to make the larger sphere(will have a diameter of 20mm or radius of 1cm)

Let's say there are N smaller spheres needed to make the larger sphere(their volumes must be the same)

\(N*\frac{4}{3}*\pi*1^3 = \frac{4}{3}*\pi*20^3\) ->\(N = 20*20*20 = 8000\)

Therefore, 8000(Option D) cubes are needed to make the large sphere(which covers maximum volume in the cuboid)
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A sphere is to be formed by melting smaller sphere cubes of diameter 17 Jun 2018, 12:34
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Solution



Given:
    • A sphere is to be formed by melting smaller spheres of diameter 20 mm
      o Radius = 10 mm = 1 cm
    • The sphere needs to be placed inside a cuboid of length 60 cm, width 50 cm, and height 40 cm, such that it covers the maximum possible volume

To find:
    • How many smaller spheres are required

Approach and Working:
If the bigger sphere is to cover the maximum possible volume of the cuboid, its diameter should be equal to least of (60, 50, 40) = 40 cm [because it must be placed inside the cuboid]

    • Therefore, radius of the bigger sphere = 20 cm

If we assume that n number of smaller spheres are needed, then we can write
    • Volume of the bigger sphere = n * volume of each small sphere
    Or, \(\frac{4}{3} * π * 20^3 = n * \frac{4}{3} * π * 1^3\)
    Or, \(n = \frac{20^3}{1^3} = 8000\)

Hence, the correct answer is option D.

Answer: D
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A sphere is to be formed by melting smaller sphere cubes of diameter 05 Jul 2019, 06:14
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A sphere is to be formed by melting smaller sphere cubes of diameter 20 mm. The sphere needs to be placed inside a cuboid of length 60 cm, width 50 cm, and height 40 cm such that it covers the maximum possible volume. How many smaller cubes are needed?

    A. 8
    B. 80
    C. 800
    D. 8000
    E. 80000


Articles on Geometry

Common Mistakes in Geometry Questions
Mastering Important Concepts tested by GMAT in Triangles - I
Mastering Important Concepts tested by GMAT in Triangles - II
Mastering Important Concepts tested by GMAT in Triangles - III

Consolidated list of all the articles

Must Read Articles and Practice Questions to score Q51 !!!!

Let R and r are the radii of Big sphere and small sphere cubes respectively.

Given r=d/2=20mm/2=10mm=1cm

Given, The Big sphere needs to be placed inside a cuboid of length 60 cm, width 50 cm, and height 40 cm such that it covers the maximum possible volume. So this to happen, height of the given cuboid must be same as diameter of big sphere

Hence, Diameter of big sphere=40cm
Or, R=40/2=20cm
Given,A sphere is to be formed by melting smaller sphere cubes of diameter 20 mm.----(1)
So, r=20mm/2=10mm=1cm
Let 'n' be the no of small spherical cubes.
from (1), we have Volume of big sphere=n *Volume of one small sphere cube
Or,\(\frac{4}{3}π*R^3=n*\frac{4}{3}π*r^3\)
Or, \(\frac{4}{3}π*{20}^3=n*\frac{4}{3}π*1^3\)
Or, n=\(20^3\)=8000 nos

Ans. (D)
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Could you please explain why you have used the diameter to calculate the volume of a sphere instead of the radius?
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A sphere is to be formed by melting smaller sphere cubes of diameter 05 Jul 2019, 09:03
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[/quote]Could you please explain why you have used the diameter to calculate the volume of a sphere instead of the radius?[/quote]


Given, The Big sphere needs to be placed inside a cuboid of length 60 cm, width 50 cm, and height 40 cm such that it covers the maximum possible volume. So this to happen, height of the given cuboid must be same as diameter of big sphere.
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A sphere is to be formed by melting smaller sphere cubes of diameter 16 Sep 2019, 00:08
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A sphere is to be formed by melting smaller sphere cubes of diameter 20 mm. The sphere needs to be placed inside a cuboid of length 60 cm, width 50 cm, and height 40 cm such that it covers the maximum possible volume. How many smaller cubes are needed?

    A. 8
    B. 80
    C. 800
    D. 8000
    E. 80000


Articles on Geometry

Common Mistakes in Geometry Questions
Mastering Important Concepts tested by GMAT in Triangles - I
Mastering Important Concepts tested by GMAT in Triangles - II
Mastering Important Concepts tested by GMAT in Triangles - III

Consolidated list of all the articles

Must Read Articles and Practice Questions to score Q51 !!!!
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Given:
1. A sphere is to be formed by melting smaller sphere cubes of diameter 20 mm.
2. The sphere needs to be placed inside a cuboid of length 60 cm, width 50 cm, and height 40 cm such that it covers the maximum possible volume.

Asked: How many smaller cubes are needed?

Diameter of larger sphere = 40 cm
Volume of larger sphere = \(4/3 \pi (200)^3 mm^3 = 32000000 \pi /3 mm^3\)

Volume of smaller sphere =\(4/3 \pi 10^3 =4000/3 mm^3\)

Number of smaller spheres needed = \(\frac{\frac{32000000\pi}{3} }{ \frac{4000\pi}{3}} = 8000\)

IMO D
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A sphere is to be formed by melting smaller sphere cubes of diameter 26 Jul 2022, 09:14
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