The entire surface of a solid cube with a length of 6 inches is painted. The cube is then cut into cubes each with edge of length 1 inch. How many smaller cubes are there? How many of the smaller cubes have paint on exactly 1 face?
First, you need to visualize the cube and you will notice that it is a 6x6 cube. First step is to think of the side of the cube. It is 6x6 inches so that means that its surface area will be 36 inches and in this scenario, it means that there will be 36 cubes on the 2D shape.Now keep the number in mind. Now go back to the 3D shape and you will notice that it is basically the 2D shape we were talking about, expect there are 6 6x6 rows instead of one. Now that easy right? We just have to multiply that 36 by 6 and we get 216. Now we know that there are 216 small cubes in the bigger cube. Sorry if this doesn't help, you should copy the question word by word from your textbook or workbook.
To find the number of smaller cubes, we need to determine how many cubes can fit along each side of the larger cube. Since the length of the larger cube is 6 inches, and the length of each smaller cube is 1 inch, we can divide the length of the larger cube by the length of the smaller cube to find the number of smaller cubes along one side.
The number of smaller cubes along one side is given by:
Number of smaller cubes = Length of larger cube / Length of smaller cube
Number of smaller cubes = 6 inches / 1 inch = 6 cubes
Since there are 6 cubes along one side, the total number of smaller cubes is the cube of this value:
Total number of smaller cubes = (Number of smaller cubes)³
Total number of smaller cubes = 6 cubes * 6 cubes * 6 cubes = 216 cubes
Therefore, there are 216 smaller cubes in total.
To determine how many of the smaller cubes have paint on exactly 1 face, we need to consider the cubes along the surface of the larger cube. Since the larger cube is painted on all sides, the smaller cubes on the surface will have paint on exactly 1 face, while the cubes in the interior will have paint on multiple faces.
To find the number of smaller cubes on the surface, we need to subtract the smaller cubes in the interior from the total number of smaller cubes.
The number of smaller cubes in the interior is given by:
Number of smaller cubes in the interior = (Number of smaller cubes along one side - 2)³
Number of smaller cubes in the interior = (6 - 2)³ = 4³ = 64 cubes
Therefore, the number of smaller cubes on the surface is:
Number of smaller cubes on the surface = Total number of smaller cubes - Number of smaller cubes in the interior
Number of smaller cubes on the surface = 216 cubes - 64 cubes = 152 cubes
Hence, there are 152 smaller cubes that have paint on exactly 1 face.