A solid wooden cube is painted blue on the outside. The cube is then cut into 27 smaller cubes
of equal size. What fraction of the total surface area of these new cubes is blue?
To determine the fraction of the total surface area of the smaller cubes that is blue, we need to consider the number of blue faces and the total number of faces in each cube.
First, we need to determine the number of smaller cubes that have at least one blue face. Since the original cube is cut into 27 smaller cubes, each side of the original cube is divided into 3 smaller cubes. Therefore, there will be 9 smaller cubes on each face of the original cube.
Next, we need to calculate the total number of faces in each cube. A cube has 6 faces, so each of the 27 smaller cubes has 6 faces.
Now, let's calculate the number of blue faces. Since the original cube is completely blue on the outside, each face (including the inner faces) of the original cube is blue. Therefore, each smaller cube that was cut from a blue face of the original cube will also have a blue face.
Since there are 9 smaller cubes on each face of the original cube, and there are 6 faces on the original cube, there are a total of 6 * 9 = 54 smaller cubes that have at least one blue face.
Now we can calculate the fraction of the total surface area that is blue:
The total number of faces in the 27 smaller cubes is 27 * 6 = 162 faces.
The total number of blue faces is 54.
Therefore, the fraction of the total surface area of the smaller cubes that is blue is 54/162, which simplifies to 1/3.
So, 1/3 of the total surface area of the smaller cubes is blue.
To find the fraction of the total surface area of the 27 smaller cubes that is blue, we need to determine the number of smaller cubes that have at least one blue face.
When we cut the original cube into 27 smaller cubes, each small cube will have 6 faces. Since the cube was painted blue on the outside, only the faces on the outside of the larger cube will be blue.
The larger cube has 6 faces, and since it was painted blue on the outside, all of its faces are blue. So, each of the 27 smaller cubes will have at least one blue face.
Therefore, the fraction of the total surface area of the smaller cubes that is blue is 27/27, which simplifies to 1/1 or 1.
So, the entire surface area of the smaller cubes is blue.
let's say that the cube has an edge length of 3 units
so the blue surface area is
... 6 * 3^2
each of the smaller cubes has an edge length of 1 unit
so their collective surface area is
... 27 * 6 * 1^2