A big cube is formed by rearranging the 160 coloured and 56 non-coloured similar cubes in such a way that the exposure of the coloured cubes to the outside is minimum. The percentage of exposed area to the coloured is :
A. 25.9%
B. 44.44%
C. 35%
D. 61%
E. None of these
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A big cube is formed by rearranging the 160 coloured and 56
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prab.sahi01
- Junior | Next Rank: 30 Posts
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prab.sahi01
- Junior | Next Rank: 30 Posts
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- Joined: Sat Oct 20, 2018 4:55 am
prab.sahi01 wrote:A big cube is formed by rearranging the 160 coloured and 56 non-coloured similar cubes in such a way that the exposure of the coloured cubes to the outside is minimum. The percentage of exposed area to the coloured is :
A. 25.9%
B. 44.44%
C. 35%
D. 61%
E. None of these
Can someone help me with this ??
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arihantdas
- Newbie | Next Rank: 10 Posts
- Posts: 7
- Joined: Wed Dec 19, 2018 12:03 am
Minimising the surface area of coloured cubes means maximising the surface area of uncoloured cubes in the big cube.
Since all cubes are similar, total cubes is 160+56=216 and if we take the length of cube as 1 unit,then total surface area will be 6x (length)xlength=216 units
Now in any big cube, maximum surface area exposed will be at 8 corners where 3 surfaces of small cubes will be exposed and then 2 surfaces of all small cubes along all the edges of the big cube will be exposed thereafter. this gives total surface area as
8 corners of big cube x 3 unit area (surfaces) of small cube at each corner + 2 unit area (surfaces) of small cube x 4 such small cubes at each edge of big cube
= 8 X 3 + 2 x 4 x 12 (total edges of a cube is 12)
=120 (max surface area of uncoloured cubes)
So total surface area of coloured cubes will be 216-120= 96
and hence the percentage will be 96/216= 44.44%
Since all cubes are similar, total cubes is 160+56=216 and if we take the length of cube as 1 unit,then total surface area will be 6x (length)xlength=216 units
Now in any big cube, maximum surface area exposed will be at 8 corners where 3 surfaces of small cubes will be exposed and then 2 surfaces of all small cubes along all the edges of the big cube will be exposed thereafter. this gives total surface area as
8 corners of big cube x 3 unit area (surfaces) of small cube at each corner + 2 unit area (surfaces) of small cube x 4 such small cubes at each edge of big cube
= 8 X 3 + 2 x 4 x 12 (total edges of a cube is 12)
=120 (max surface area of uncoloured cubes)
So total surface area of coloured cubes will be 216-120= 96
and hence the percentage will be 96/216= 44.44%
