(a) A cube is painted red on all six sides, then cut into 125 smaller congruent cubes.
How many of the small cubes are painted red on exactly one side?
(b) A cube is painted red on all six sides, then cut into 125 smaller congruent cubes.
How many of the small cubes are not painted red on any sides?
(c) A cube is painted red on all six sides, then cut into 125 smaller congruent cubes.
How many of the small cubes are not painted red on any sides?
I really do not understand this problem.
Bruh, stop cheating. It's stupid.
Ploop you guys.
No problem! Let's break down the problem and find the answers step by step.
(a) To determine how many small cubes are painted red on exactly one side, let's visualize the process. Start with a large cube that has six faces, and paint each face red. When the cube is cut into smaller congruent cubes, we know that each side of the large cube will have 5 small cubes attached to it.
To get the number of small cubes painted red on exactly one side, we need to focus on the cubes on the surface of the large cube. Since each face has 5 small cubes attached to it, and there are 6 faces, we have a total of 6 * 5 = 30 small cubes on the outer surface. However, each corner of the large cube has 3 faces, so we have counted these small cubes more than once.
There are 8 corners in a cube, and each corner is counted in 3 faces (since it is shared by 3 faces). Therefore, we need to subtract the number of corners counted more than once from the total count of small cubes on the outer surface.
Since each corner is counted 3 times, we have 8 * 3 = 24 extra small cubes counted. So, the number of small cubes painted red on exactly one side is 30 - 24 = 6.
(b) To find the number of small cubes not painted red on any sides, we need to consider the cubes in the center of the large cube and hidden from view. Since the large cube is made up of 125 smaller cubes, and we know that there are 6 small cubes attached to each face, we can calculate the total number of small cubes that are not painted red on any sides by subtracting the visible cubes from the total number of cubes.
The total number of small cubes in the large cube is 125, and the number of visible small cubes is 30 (as we found in part (a)). Therefore, the number of small cubes not painted red on any sides is 125 - 30 = 95.
(c) It seems that part (c) is the same as part (b). So, the number of small cubes not painted red on any sides is still 95.
I hope this explanation helps you understand the problem better! Let me know if you have any further questions.