beyondgmatscore wrote:
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.
A very good explanation indeed. I tried a different approach and ended up with incorrect answer. Please help me correct with my mistake here.
Let us consider that there are five individual small cubes that make up the edge of the cube. I have added one picture for representation.
Assume that the cube is made up of 5 small individual cubes as shown in the picture. This is only one face of the 6 faces of the cube. We consider the outer cubes as below:
Blue Cubes - The 3X3=9 blue faces have no corners common and repeat at each of the faces of the cube. Thus, there are in total 3X3X6 = 54 blue cubes in total
Green Cubes - We have 12 (3X4) Green cubes on one face that share their edges with other cubes. We will have similar 12 Green cubes on the other side. Thus, we have 24 Green cubes.
Orange Cubes - We have 4 such cubes that are at corners and share their edges with cubes at other sides. We will have 4X2=8 such orange cubes.
Total cubes facing the air = Blue + Green + Orange = 54+24+8=86
Please tell me my mistake. Thanks.
Attachments
cube.JPG [ 5.07 KiB | Viewed 60437 times ]