(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.
A small cube is selected at random and rolled. What is the probability that it has a red side facing up?
To solve these questions, we can visualize the process and break it down step by step.
a) To find the number of small cubes painted red on exactly one side, we need to determine the number of cubes that have exactly one face painted red. Since each edge of the big cube is divided into 5 segments (4 internal and 1 external), we can conclude that there are 5 cubes along each edge of the larger cube. Thus, the total number of small cubes is 5 x 5 x 5 = 125.
Now, let's count the cubes with exactly one red side. Each edge of the larger cube has 4 cubes with one face painted red (excluding the corner cubes). Since there are 12 edges in total, there are 12 x 4 = 48 cubes with exactly one red side.
b) To determine the number of small cubes that are not painted red on any sides, we need to find the cubes that do not have any red faces. Since each edge of the big cube consists of 5 cubes, except for the 3 x 3 x 3 inner cubes, which have 3 painted faces, we can conclude that there are 5 x 5 x 5 - 27 = 98 cubes with no red sides.
c) To find the probability of rolling a small cube with a red side facing up, we need to know the total number of cubes and the number of cubes with at least one red face. Since there are 125 small cubes in total, and we found in part b that there are 98 cubes without any red sides, we can calculate the probability as follows:
Probability = Number of favorable outcomes / Total number of outcomes
Probability = (Number of cubes with at least one red side) / (Total number of cubes)
Probability = (125 - 98) / 125
Probability = 27 / 125
Probability ≈ 0.216 or 21.6%