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02 Mar 2011, 03:23
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C
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62% (01:27) correct
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A large cube consists of 125 identical small cubes, how many of the small cubes are exposed in air?
(A) 64
(B) 72
(C) 98
(D) 100
(E) 116
(A) 64
(B) 72
(C) 98
(D) 100
(E) 116
02 Mar 2011, 03:29
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banksy
As the cube consists of 125=5^3 identical small cubes then its sides are built with 5 small cubes. The sides of the inner cube are built with 3 small cube (5 - 2 small edge cubes), thus there are 125-3^3=98 cubes exposed.
Answer: C.
02 Mar 2011, 23:31
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OK - let me try and explain this.
The question says that a large cube is actually made up of 125 smaller cubes.
Since 125 = \(5^3\), it implies that each side of the large cube is actually made up of 5 cubes. For simplicity, if large cube has side 5 cm, then small cube has side one cm and hence large cube with volume 125 cu. cm. is actually 125 cubes of 1 cu. cm. each.
Now, the question is how many cubes are exposed, which simply means that how many of such small cubes constitute outer surface (which is exposed to air)?
So, we can work this out by finding those cubes that do not have any of the surfaces exposed to air.
Lets visualise the cube as 5 squares of 25 cubes each placed over one another, Now top and bottom square are clearly the ones with all cubes exposed to air.
For squares 2, 3 and 4, 16 cubes that make the four sides are exposed to air, but 9 cubes that are enclosed within these sides are not exposed. So total of 9*3 = 27 cubes are not exposed to air.
Hence, 125 - 27 = 98 cubes are exposed to air.
You can quickly visualise that except a 3x3 cube within the larger cube, everything else is exposed, so you can quickly do \(5^3\) - \(3^3\) and derive the answer, as Bunuel has done.
The question says that a large cube is actually made up of 125 smaller cubes.
Since 125 = \(5^3\), it implies that each side of the large cube is actually made up of 5 cubes. For simplicity, if large cube has side 5 cm, then small cube has side one cm and hence large cube with volume 125 cu. cm. is actually 125 cubes of 1 cu. cm. each.
Now, the question is how many cubes are exposed, which simply means that how many of such small cubes constitute outer surface (which is exposed to air)?
So, we can work this out by finding those cubes that do not have any of the surfaces exposed to air.
Lets visualise the cube as 5 squares of 25 cubes each placed over one another, Now top and bottom square are clearly the ones with all cubes exposed to air.
For squares 2, 3 and 4, 16 cubes that make the four sides are exposed to air, but 9 cubes that are enclosed within these sides are not exposed. So total of 9*3 = 27 cubes are not exposed to air.
Hence, 125 - 27 = 98 cubes are exposed to air.
You can quickly visualise that except a 3x3 cube within the larger cube, everything else is exposed, so you can quickly do \(5^3\) - \(3^3\) and derive the answer, as Bunuel has done.
