Question

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?

Answer 1

The volume of cube C is

`V=192√(3)\ units^3`

Explanation:

step 1

Find the diameter of sphere B

we know that

When a cube is inscribed in a sphere, the long diagonal of the cube is a diameter of the sphere

Let

L ----> the length side of cube A

d ----> the diagonal of the base of cube A

D ---> the long diagonal of cube A

Find the diagonal of the base of cube A

Applying the Pythagorean Theorem

`d^2=L^2+L^2`

we have

`L=4\ units`

substitute

`d^2=4^2+4^2\\d^2=32\\d=4√(2)\ units`

Find the long diagonal of cube A

Applying the Pythagorean Theorem

`D^2=d^2+L^2`

substitute

`D^2=32+4^2\\D^2=48\\D=4√(3)\ units`

step 2

we know that

If sphere B is inscribed in cube C, then the length side of cube C is equal to the diameter of sphere B

Let

c ----> the length side of cube C

we have that

`c=4√(3)\ units`

The volume of cube C is equal to

`V=c^3`

substitute

`V=(4√(3))^3`

`V=192√(3)\ units^3`