Cube A is inscribed in sphere B, which is inscribed in cube C. If the sides of cube A have length 4, what is the volume of cube C?

the volume of a sphere inscribed in a cube of diagonal 4√3 is

To find the volume of cube C, we need to determine the length of its sides.

First, let's find the diameter of sphere B. The diameter of a sphere is equal to twice the radius. Since cube A is inscribed in sphere B, the diameter of sphere B is equal to the diagonal of cube A.

The diagonal of a cube can be found using the Pythagorean theorem. In this case, the sides of cube A have length 4, so the diagonal can be calculated as follows:

diagonal = √(side^2 + side^2 + side^2)
= √(4^2 + 4^2 + 4^2)
= √(16 + 16 + 16)
= √(48)
= 4√3

Now that we have the diameter of sphere B, we can calculate its radius by dividing the diameter by 2:

radius = diameter / 2
= (4√3) / 2
= 2√3

Next, we can find the length of the sides of cube C by doubling the radius of sphere B:

side of cube C = 2 * radius
= 2 * 2√3
= 4√3

Finally, we can calculate the volume of cube C by cubing the length of its sides:

volume of cube C = (side of cube C)^3
= (4√3)^3
= 4^3 * (√3)^3
= 64 * 3√3
= 192√3

Therefore, the volume of cube C is 192√3.

When a cube is inscribed in a sphere, the long diagonal of the cube is a diameter of the sphere. So, the sphere has diameter 4√3

Similarly, the diameter of the sphere is equal to the side of cube C, so it has side 4√3.

so, ...