a cube of side 4 cm is cut into i cmcubes. the ratio of the surface areas of the original cubes and cut-out cubes is
SA of original cube = 6(4^2) = 96 cm^2
I will assume it will be cut into cubes, each one i cm on a side.
So number of such cubes = 4^3 / i^3
surface area of each small cube = 6i^2
total surface area of all the little cubes
= (4^3)/(i^3)*(6i^2)
= 64(6)/i = 384/i
so ratio of SA of original : SA of all little ones
= 96 : 384/i
= 1 : 4/i
= i : 4
If you meant that there will be i cubes to be cut out,
the calculations will require cube roots and square roots
let me know
To find the ratio of the surface areas of the original cube and the cut-out cubes, we need to determine the number of cut-out cubes first.
The original cube has a side length of 4 cm. If we cut it into smaller cubes with a side length of i cm, we can calculate the number of cut-out cubes by dividing the side length of the original cube by the side length of the cut-out cube.
Number of cut-out cubes = (Side length of original cube) / (Side length of cut-out cube)
Number of cut-out cubes = 4 cm / i cm = 4/i
Now that we know the number of cut-out cubes, we can find the surface area of the original cube and the surface area of one cut-out cube.
The surface area of the original cube is given by:
Surface area of original cube = 6 * (Side length of original cube)^2
Surface area of original cube = 6 * (4 cm)^2 = 6 * 16 cm^2 = 96 cm^2
The surface area of one cut-out cube is given by:
Surface area of one cut-out cube = 6 * (Side length of cut-out cube)^2
Surface area of one cut-out cube = 6 * (i cm)^2 = 6i^2 cm^2
Now, we can find the ratio of the surface areas by dividing the surface area of the original cube by the total surface area of the cut-out cubes.
Ratio = (Surface area of original cube) / (Surface area of one cut-out cube * Number of cut-out cubes)
Ratio = 96 cm^2 / (6i^2 cm^2 * (4/i))
Ratio = 16 / i
Therefore, the ratio of the surface areas of the original cube and the cut-out cubes is 16/i.