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

Answer:

The volume of cube C is

$V=192\sqrt{3}\ units^3$

Step-by-step 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

$d^2=4^2+4^2\\d^2=32\\d=4\sqrt{2}\ units$

Find the long diagonal of cube A

Applying the Pythagorean Theorem

$D^2=d^2+L^2$

substitute

$D^2=32+4^2\\D^2=48\\D=4\sqrt{3}\ units$

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

$c=4\sqrt{3}\ units$

The volume of cube C is equal to

$V=c^3$

substitute

$V=(4\sqrt{3})^3$

$V=192\sqrt{3}\ units^3$