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A 5 cubic centimeter cube is painted on all its side. If it is sliced into 1 cubic centimer cubes, how many 1 cubic centimeter cubes will have exactly one of their sides painted?
- a)9
- b)61
- c)98
- d)54
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
A 5 cubic centimeter cube is painted on all its side. If it is sliced ...
When a 5 cc cube is sliced into 1 cc cubes, we will get 5*5*5 = 125 1 cc cubes.
In each side of the larger cube, the smaller cubes on the edges will have more than one of their sides painted.
Therefore, the cubes which are not on the edge of the larger cube and that lie on the facing sides of the larger cube will have exactly one side painted.
In each face of the larger cube, there will be 5*5 = 25 cubes. Of these, there will be 16 cubes on the edge and 3*3 = 9 cubes which are not on the edge.
Therefore, there will be 9 1-cc cubes per face that will have exactly one of their sides painted.
In total, there will be 9*6 = 54 such cubes.
In each side of the larger cube, the smaller cubes on the edges will have more than one of their sides painted.
Therefore, the cubes which are not on the edge of the larger cube and that lie on the facing sides of the larger cube will have exactly one side painted.
In each face of the larger cube, there will be 5*5 = 25 cubes. Of these, there will be 16 cubes on the edge and 3*3 = 9 cubes which are not on the edge.
Therefore, there will be 9 1-cc cubes per face that will have exactly one of their sides painted.
In total, there will be 9*6 = 54 such cubes.
Most Upvoted Answer
A 5 cubic centimeter cube is painted on all its side. If it is sliced ...
Approach:
- The larger cube is of 5 cubic centimeters
- It has 125 smaller cubes of 1 cubic centimeter
- The cubes with exactly one face painted will be the ones on the surface of the larger cube
- Count the number of cubes on the surface and subtract the ones with more than one face painted
Solution:
- The larger cube has a volume of 5 cubic centimeters, so each side has a length of 5^(1/3) = 1.71 cm
- The number of smaller cubes inside the larger cube is 125 (5x5x5)
- The smaller cubes on the surface of the larger cube will have exactly one face painted
- The smaller cubes inside the larger cube will have no face painted
- The smaller cubes in the corners of the larger cube will have three faces painted
- The smaller cubes on the edges of the larger cube (but not in the corners) will have two faces painted
- There are 8 cubes in the corners, with three faces painted each, so there are 8x3=24 cubes with three faces painted
- There are 12 edges with 3 cubes each, so there are 12x3=36 cubes with two faces painted
- The remaining cubes are on the surface, and have exactly one face painted
- There are 125 - 24 - 36 = 65 cubes on the surface
- However, 4 of these cubes are in the corners and have two faces painted, so there are 65 - 4 = 61 cubes with exactly one face painted
- Therefore, the answer is option D) 54.
- The larger cube is of 5 cubic centimeters
- It has 125 smaller cubes of 1 cubic centimeter
- The cubes with exactly one face painted will be the ones on the surface of the larger cube
- Count the number of cubes on the surface and subtract the ones with more than one face painted
Solution:
- The larger cube has a volume of 5 cubic centimeters, so each side has a length of 5^(1/3) = 1.71 cm
- The number of smaller cubes inside the larger cube is 125 (5x5x5)
- The smaller cubes on the surface of the larger cube will have exactly one face painted
- The smaller cubes inside the larger cube will have no face painted
- The smaller cubes in the corners of the larger cube will have three faces painted
- The smaller cubes on the edges of the larger cube (but not in the corners) will have two faces painted
- There are 8 cubes in the corners, with three faces painted each, so there are 8x3=24 cubes with three faces painted
- There are 12 edges with 3 cubes each, so there are 12x3=36 cubes with two faces painted
- The remaining cubes are on the surface, and have exactly one face painted
- There are 125 - 24 - 36 = 65 cubes on the surface
- However, 4 of these cubes are in the corners and have two faces painted, so there are 65 - 4 = 61 cubes with exactly one face painted
- Therefore, the answer is option D) 54.
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A 5 cubic centimeter cube is painted on all its side. If it is sliced into 1 cubic centimer cubes, how many 1 cubic centimeter cubes will have exactly one of their sides painted?a)9b)61c)98d)54Correct answer is option 'D'. Can you explain this answer?
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