there are 27 white cubes assembled to forma large cube.the outside surface of the large cube is painted red. the large cube is then separated into the 27 smaller cubes.how many of the small cubes will have red paint on exactly the following number of faces?
A= three faces
B= two faces
C= one face
D= no face
pick one ___________
3 * 3 * 3 = 27, so each side of the larger cube is 3 smaller cubes. Does that help you?
I will start you out with A. Those cubes in the corners will have 3 sides painted. How many corners does the large cube have?
It might help to try to draw the situation.
To solve this problem, we need to visualize the large cube and understand how each smaller cube fits within it.
The large cube is formed by assembling 27 small cubes together. Since the large cube has 6 faces (top, bottom, front, back, left, and right), we can infer that each small cube will have one face exposed to the outside surface of the large cube.
Now let's analyze each case:
A) Three Faces:
In order for a small cube to have three faces painted red, it means that it was positioned on one of the corners of the large cube. Each corner of the large cube is formed by the intersection of 3 edges, which means there are 8 corners in total. Therefore, there are 8 small cubes that have three faces painted red.
B) Two Faces:
A small cube will have two faces painted red if it is positioned on one of the edges of the large cube but not at a corner. Each edge of the large cube is formed by the connection of 2 smaller cubes, and there are 12 edges in total. However, we need to exclude the corners counted in the previous case (A). Therefore, there are 12 - 8 = 4 small cubes that have two faces painted red.
C) One Face:
A small cube will have one face painted red if it is positioned on one of the faces of the large cube but not on an edge or corner. Each face of the large cube is composed of 9 small cubes (3x3), and there are 6 faces in total. Again, we need to exclude the edges and corners counted in the previous cases. Therefore, there are 6 x 9 - 12 - 8 = 46 small cubes that have one face painted red.
D) No Face:
Lastly, the remaining small cubes that were positioned inside the large cube will have no face painted red. To calculate this, we subtract the number of cubes with any number of painted faces from the total number of small cubes: 27 - 8 - 4 - 46 = 27 - 58 = -31. However, in this case, we cannot have a negative number of cubes, so we consider there are 0 small cubes that have no face painted red.
So the final answer is:
A) Three Faces: 8 small cubes
B) Two Faces: 4 small cubes
C) One Face: 46 small cubes
D) No Face: 0 small cubes