343 smaller but identical cubes have been put together to form a larger cube. This larger cube is now painted on all 6 faces. How many of the smaller cubes have no face painted at all?
Answers
No face painted cube at all = 125
Given:
343 smaller but identical cubes have been put together to form a larger cube.
This larger cube is now painted on all 6 faces.
To find:
How many of the smaller cubes have no face painted at all.
Formula used:
Volume of cube = (Side)³
Faces of cube = 6
Sides of cube = 12
Corner of cube = 8
Explanation:
Consider the smaller cube has 1 unit side so the volume of the cube is 343
From this information we calculate side of larger cube.
Volume of cube = (Side)³
343 = (Side)³
Side of larger cube = 7 unit
There are 6 faces in the cube
Case 1:
Now There are Eight corner cubes which are common to 3 faces
So we can subtract 8×2 = 16 cube subtract from above.
Case 2:
Now there are 12 sides are there in the cube which contain two faces common to each other so each side contain 7 cubes But we can consider 2 corner cube in above case so remaining cubes are 5 in each side For 12 side its 5×12 = 60
There are 6 faces in the cube
The total number of cube painted = 7² ×6 = 294
We can subtract the cubes in case 1 and case 2
Total number of cube painted = 294 - 16 -60 =218
No face painted at all = 343 - 218 = 125
No face painted cube at all = 125
For better understanding refer to the picture which i have attached below
To learn more...
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