A cube with an edge length of 4 in is painted on all its sides. Then the cube is cut into 64 cubes with an edge length of 1 in. What percent of the total surface area is painted?
100 in. squared or 6 in. Squared
24
To find the total surface area of the large cube, we calculate the surface area of one face and then multiply it by 6, since a cube has 6 faces:
Surface area of one face = (edge length)^2 = (4 in)^2 = 16 in^2
Total surface area of the large cube = 6 * Surface area of one face = 6 * 16 in^2 = 96 in^2
Since all the sides of the large cube are painted, the painted surface area is equal to the total surface area of the large cube, which is 96 in^2.
Next, we need to calculate the total surface area of the smaller cubes. Since there are 64 smaller cubes, we calculate the surface area of one cube and then multiply it by 64:
Surface area of one smaller cube = 6 * (edge length)^2 = 6 * (1 in)^2 = 6 in^2
Total surface area of the smaller cubes = 64 * Surface area of one smaller cube = 64 * 6 in^2 = 384 in^2
Finally, we can calculate the percentage of the total surface area that is painted:
Percentage of painted surface area = (Painted surface area / Total surface area) * 100
= (96 in^2 / 384 in^2) * 100
= 0.25 * 100
= 25%
Therefore, 25% of the total surface area is painted.
To find the percentage of the total surface area that is painted, we need to determine the total surface area of the painted cubes and divide it by the total surface area of all the smaller cubes.
The total surface area of the painted cubes can be calculated by finding the surface area of one painted cube and multiplying it by the number of cubes we have. Since each cube has a side length of 4 inches, the surface area of one cube is 6 * (4 inches)^2.
Surface area of one painted cube = 6 * (4 inches)^2 = 96 square inches.
Since the large cube is cut into 64 smaller cubes, we have 64 painted cubes.
Total surface area of painted cubes = 64 painted cubes * 96 square inches per painted cube = 6144 square inches.
Now, let's calculate the total surface area of all the smaller cubes. Since the side length of each smaller cube is 1 inch, the surface area of one cube is 6 * (1 inch)^2.
Surface area of one smaller cube = 6 * (1 inch)^2 = 6 square inches.
Since there are 64 smaller cubes, the total surface area of all the smaller cubes is 64 smaller cubes * 6 square inches per smaller cube = 384 square inches.
Finally, to find the percentage of the total surface area that is painted, we divide the total surface area of the painted cubes by the total surface area of all the smaller cubes and multiply by 100.
Percentage of total surface area painted = (Total surface area of painted cubes / Total surface area of all smaller cubes) * 100
Percentage of total surface area painted = (6144 square inches / 384 square inches) * 100
Percentage of total surface area painted = 16 * 100 = 1600%
Therefore, 1600% of the total surface area is painted.