A sphere is to be formed by melting smaller sphere cubes of diameter

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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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 \(pi * r^2\)
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my bad, was thinking about the area, sorry.

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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No idea what a sphere cube is

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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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)

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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Could you please explain why you have used the diameter to calculate the volume of a sphere instead of the radius?

[/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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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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